Trigonometric Identities: Videos & Practice Problems

Understanding the properties of trigonometric functions is essential for simplifying expressions. The cosine function is even, meaning $f(-x)=f\left(x\right)$, while the sine function is odd, so $f(-x)=-f\left(x\right)$. Pythagorean identities, such as $\sin ²\left(\theta \right)+\cos ²\left(\theta \right)=1$, help simplify squared functions. Additionally, sum and difference identities allow for the evaluation of trigonometric functions with combined angles, enhancing problem-solving skills in trigonometry.

Even and Odd Identities

Understanding the classification of trigonometric functions as even or odd is essential for simplifying expressions and solving problems in trigonometry. An even function is defined by the property that f(-x) = f(x), indicating symmetry about the y-axis. In contrast, an odd function satisfies f(-x) = -f(x), demonstrating symmetry about the origin.

Among the trigonometric functions, the cosine function is classified as even. For example, evaluating the cosine at \(\frac{\pi}{2}\) yields cos\(\left(\frac{\pi}{2}\right) = 0\), and similarly, cos\(\left(-\frac{\pi}{2}\right) = 0\). This consistency allows us to generalize that cos\(-\theta = cos\(\theta\). Thus, when simplifying expressions like cos\(-\frac{\pi}{4}\), we can conclude it equals cos\(\left(\frac{\pi}{4}\right)\), which evaluates to \(\frac{\sqrt{2}}{2}\).

On the other hand, the sine function is classified as odd. For instance, sin\(\left(\frac{\pi}{2}\right) = 1\) and sin\(-\frac{\pi}{2} = -1\), leading to the conclusion that sin\(-\theta = -sin\(\theta\). This property is useful when evaluating expressions like sin\(-\frac{\pi}{6}\), which can be rewritten as -sin\(\left(\frac{\pi}{6}\right)\), yielding - \(\frac{1}{2}\).

Furthermore, the secant function, being the reciprocal of cosine, is also even, while cosecant, the reciprocal of sine, is odd. The tangent function, defined as tan\(\theta = \frac{sin\(\theta\)}{cos\(\theta\)}, inherits the odd property from sine, making it odd as well. Consequently, cotangent is also classified as odd.

These classifications lead to the formulation of even-odd identities, which are equations that hold true for all values of their variables. For example, knowing that cos\(-\theta = cos\(\theta\) and tan\(-\theta = -tan\(\theta\) allows for simplifications in various trigonometric expressions. When faced with a negative argument in trigonometric functions, applying these identities becomes crucial for efficient problem-solving.

In summary, recognizing the even and odd nature of trigonometric functions not only aids in simplifying expressions but also enhances understanding of their graphical behavior. Mastery of these concepts is vital as one progresses through more complex trigonometric problems.

Even and Odd Identities Example 1

Understanding even-odd identities is crucial for simplifying trigonometric expressions, especially when dealing with negative arguments. These identities allow us to rewrite expressions in a more manageable form, often eliminating negative angles entirely.

For instance, consider the expression involving the negative tangent of negative theta. According to the even-odd identity for tangent, we know that:

tan(-θ) = -tan(θ)

Thus, we can rewrite the expression as follows:

-tan(-θ) = -(-tan(θ)) = tan(θ)

This transformation shows that two negatives result in a positive, leading us to a simplified expression of just tan(θ).

Next, let’s analyze the expression sin(-θ) / cos(-θ). Here, we apply the even-odd identities for sine and cosine:

sin(-θ) = -sin(θ) and cos(-θ) = cos(θ).

Substituting these identities into the expression gives us:

sin(-θ) / cos(-θ) = -sin(θ) / cos(θ)

Recognizing that sin(θ) / cos(θ) is equivalent to tan(θ), we can further simplify this to:

-tan(θ).

Alternatively, one could directly recognize that sin(-θ) / cos(-θ) simplifies to tan(-θ), which again leads us to:

tan(-θ) = -tan(θ).

Both methods yield the same result, demonstrating that there are often multiple pathways to arrive at the correct simplification. Mastering these identities and their applications will enhance your ability to manipulate trigonometric expressions effectively.

Use the even-odd identities to evaluate the expression.
$\cos\left(-\theta\right)-\cos\theta$

• A. 0
• B. $-\cos\theta$
• C. $2\cos\theta$
• D. $-2\cos\theta$

Answer: (A)

Use the even-odd identities to evaluate the expression.
$-\cot\left(\theta\right)\cdot\sin\left(-\theta\right)$

• A. $\tan\theta$
• B. $-\cos\theta$
• C. $\cos\theta$
• D. $\frac{\cos\theta}{\sin^2\theta}$

Answer: (C)

Select the expression with the same value as the given expression.
$\sec\left(-\frac{4\pi}{5}\right)$

• A. $\cos\left(\frac{4\pi}{5}\right)$
• B. $-\cos\left(\frac{4\pi}{5}\right)$
• C. $\sec\left(\frac{4\pi}{5}\right)$
• D. $-\sec\left(\frac{4\pi}{5}\right)$

Answer: (C)

Select the expression with the same value as the given expression.
$\sin\left(-38\degree\right)$

• A. $\sin38\degree$
• B. $-\sin38\degree$
• C. $-\sin\left(-38\degree\right)$
• D. $\frac{1}{-\sin38\degree}$

Answer: (B)

Pythagorean Identities

When working with trigonometric expressions, you may encounter forms that include squared sine and cosine functions, such as ${\sin }^2\left(\frac{11}{6}\right)+{\cos }^2\left(\frac{11}{6}\right).$ This expression can seem daunting at first, but it simplifies to one, thanks to the Pythagorean identities. These identities are derived from the Pythagorean Theorem, which states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: $a^2+b^2=c^2.$ By applying this theorem to the unit circle, where the hypotenuse is 1, we find that ${\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)=\; 1.$ This fundamental identity holds true for any angle.

To derive additional Pythagorean identities, you can manipulate the first identity. Dividing by $\cos (\mathrm{\theta <mo>)</mo>}$ yields $\left(1+\csc\theta\right)\left(1-\csc\theta\right)$

In this problem, we aim to rewrite the expression \(\frac{1}{1 + \cos(\theta)}\) without a fraction by utilizing Pythagorean identities. Initially, the expression does not contain any squared trigonometric functions, which makes it less straightforward to apply these identities. However, we can manipulate the expression to achieve our goal.

To begin, we multiply both the numerator and the denominator by \(1 - \cos(\theta)\). This step is crucial as it maintains the value of the expression while allowing us to transform the denominator. The multiplication yields:

Numerator: \(1 \cdot (1 - \cos(\theta)) = 1 - \cos(\theta)\)

Denominator: \((1 + \cos(\theta))(1 - \cos(\theta)) = 1 - \cos^2(\theta)\)

Now, we recognize that \(1 - \cos^2(\theta)\) can be rewritten using the first Pythagorean identity, which states that:

\(1 = \sin^2(\theta) + \cos^2(\theta)\)

By rearranging this identity, we find that:

\(1 - \cos^2(\theta) = \sin^2(\theta)\)

Substituting this back into our expression, we have:

\(\frac{1 - \cos(\theta)}{\sin^2(\theta)}\)

To eliminate the fraction, we can express this as:

\(\frac{1}{\sin^2(\theta)}(1 - \cos(\theta))\)

Recognizing that \(\frac{1}{\sin(\theta)}\) is the cosecant function, we can rewrite \(\frac{1}{\sin^2(\theta)}\) as \(\csc^2(\theta)\). Thus, our expression simplifies to:

\(\csc^2(\theta)(1 - \cos(\theta))\)

In conclusion, we have successfully rewritten the original expression without a fraction, resulting in:

\(\frac{1}{1 + \cos(\theta)} = \csc^2(\theta)(1 - \cos(\theta))\)

In trigonometry, sum and difference identities are essential tools for simplifying expressions involving multiple angles. These identities allow us to break down complex trigonometric functions into simpler components, making it easier to evaluate them without a calculator, even for angles not directly on the unit circle.

For instance, consider the expression $\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\; +\; \backslash frac\{\backslash pi\}\{6\}\backslash right)$. To simplify this, we can apply the sum identity for sine, which states:

$\backslash sin(a\; +\; b)\; =\; \backslash sin(a)\; \backslash cos(b)\; +\; \backslash cos(a)\; \backslash sin(b)$

Here, $a\; =\; \backslash frac\{\backslash pi\}\{2\}$ and $b\; =\; \backslash frac\{\backslash pi\}\{6\}$. Substituting these values into the identity gives:

$\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)\; +\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; \backslash sin\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)$

From the unit circle, we know that $\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; 1$, $\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; 0$, $\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$, and $\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)\; =\; \backslash frac\{1\}\{2\}$. Thus, the expression simplifies to:

$1\; \backslash cdot\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}\; +\; 0\; \backslash cdot\; \backslash frac\{1\}\{2\}\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$

This demonstrates how the sum identity can simplify the evaluation of trigonometric functions. Similarly, the difference identity for sine is:

$\backslash sin(a\; -\; b)\; =\; \backslash sin(a)\; \backslash cos(b)\; -\; \backslash cos(a)\; \backslash sin(b)$

For cosine, the identities are slightly different. The sum identity is:

$\backslash cos(a\; +\; b)\; =\; \backslash cos(a)\; \backslash cos(b)\; -\; \backslash sin(a)\; \backslash sin(b)$

And the difference identity is:

$\backslash cos(a\; -\; b)\; =\; \backslash cos(a)\; \backslash cos(b)\; +\; \backslash sin(a)\; \backslash sin(b)$

These identities are particularly useful when dealing with angles that are not standard on the unit circle, such as 15 degrees. To find $\backslash cos(15^\backslash circ)$, we can express it as $\backslash cos(45^\backslash circ\; -\; 30^\backslash circ)$. Applying the cosine difference identity yields:

$\backslash cos(45^\backslash circ)\; \backslash cos(30^\backslash circ)\; +\; \backslash sin(45^\backslash circ)\; \backslash sin(30^\backslash circ)$

Substituting known values from the unit circle gives:

$\backslash frac\{\backslash sqrt\{2\}\}\{2\}\; \backslash cdot\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}\; +\; \backslash frac\{\backslash sqrt\{2\}\}\{2\}\; \backslash cdot\; \backslash frac\{1\}\{2\}\; =\; \backslash frac\{\backslash sqrt\{6\}\}\{4\}\; +\; \backslash frac\{\backslash sqrt\{2\}\}\{4\}\; =\; \backslash frac\{\backslash sqrt\{6\}\; +\; \backslash sqrt\{2\}\}\{4\}$

In summary, sum and difference identities are powerful tools in trigonometry that allow for the simplification and evaluation of trigonometric functions involving multiple angles. By recognizing when to apply these identities, particularly with angles that can be expressed as sums or differences of known values, students can effectively tackle a wide range of problems in trigonometry.

In trigonometry, rewriting angles as the sum or difference of known angles on the unit circle can simplify the process of finding exact values for trigonometric functions. This technique is particularly useful for angles that are not commonly memorized, such as 75 degrees.

To illustrate this, consider the angle of 75 degrees. We can express it as the sum of two angles from the first quadrant of the unit circle. Notably, 30 degrees and 45 degrees can be combined since:

30° + 45° = 75°.

Using the sine sum formula, we can find:

sin(75°) = sin(30° + 45°) = sin(30°)cos(45°) + cos(30°)sin(45°).

Next, let's examine the cosine of -15 degrees. Here, we can use the angles 30 degrees and 45 degrees again. By subtracting, we find:

30° - 45° = -15°.

Thus, we can express this as:

cos(-15°) = cos(30° - 45°) = cos(30°)cos(45°) + sin(30°)sin(45°).

For angles in radians, such as 7π/12, we can also apply this method. The angles π/6, π/4, and π/3 are relevant here. To find a combination that sums to 7π/12, we can break it down as:

3π/12 + 4π/12 = 7π/12.

When simplified, this corresponds to:

π/4 + π/3.

Thus, we can express the cosine of 7π/12 as:

cos(7π/12) = cos(π/4 + π/3) = cos(π/4)cos(π/3) - sin(π/4)sin(π/3).

In summary, recognizing angles that are multiples of 15 degrees or π/12 radians allows us to effectively use sum and difference identities. This approach not only streamlines calculations but also enhances understanding of trigonometric relationships on the unit circle.

To find the exact value of the expression sin(10°) * cos(20°) + sin(20°) * cos(10°), we can utilize the sine sum formula. This expression resembles the sine addition formula, which states:

sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b)

In our case, we can identify a as 10 degrees and b as 20 degrees. Therefore, we can rewrite the expression as:

sin(10° + 20°)

Calculating the sum gives us:

sin(30°)

From the unit circle, we know that:

sin(30°) = 1/2

Thus, the exact value of the original expression is:

1/2

This approach highlights the importance of recognizing patterns in trigonometric identities, particularly the sine addition formula, to simplify complex expressions effectively.

In trigonometry, sum and difference identities are essential tools for simplifying expressions involving angles. These identities allow us to expand and simplify trigonometric functions that include variables. For instance, when expanding the expression for the sine of an angle plus a specific degree, we can apply the sum formula for sine.

Consider the expression sin(β + 30°). Using the sum formula, we can expand this to:

\[\sin(β + 30°) = \sin(β) \cos(30°) + \cos(β) \sin(30°)\]

From the unit circle, we know that cos(30°) = \frac{\sqrt{3}}{2} and sin(30°) = \frac{1}{2}. Substituting these values into the equation gives:

\[\sin(β + 30°) = \sin(β) \cdot \frac{\sqrt{3}}{2} + \cos(β) \cdot \frac{1}{2}\]

To simplify further, we can rearrange the terms to factor out the common denominator:

\[\sin(β + 30°) = \frac{1}{2} \left( \sqrt{3} \sin(β) + \cos(β) \right)\]

This expression is now simplified using the sum formula.

Next, let's explore the cosine of an angle minus a variable, such as cos(π/4 - θ). Here, we apply the difference formula for cosine:

\[\cos(π/4 - θ) = \cos(π/4) \cos(θ) + \sin(π/4) \sin(θ)\]

Again, from the unit circle, we find that both cos(π/4) and sin(π/4) equal \frac{\sqrt{2}}{2}. Substituting these values results in:

\[\cos(π/4 - θ) = \frac{\sqrt{2}}{2} \cos(θ) + \frac{\sqrt{2}}{2} \sin(θ)\]

Since both terms share the same coefficient, we can factor it out:

\[\cos(π/4 - θ) = \frac{\sqrt{2}}{2} \left( \cos(θ) + \sin(θ) \right)\]

This expression is now fully expanded and simplified using the difference formula.

By mastering these identities, you can effectively simplify a wide range of trigonometric expressions involving variables, enhancing your problem-solving skills in trigonometry.

Understanding trigonometric identities is essential for simplifying trigonometric expressions and solving problems efficiently. Among these identities, double angle identities are particularly useful. They are derived from the sum formulas for sine, cosine, and tangent, specifically when the same angle is used twice.

The double angle identity for sine states that:

$$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$$

For cosine, the double angle identity can be expressed in three forms:

$$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$$

Alternatively, using the Pythagorean identity, it can also be written as:

$$\cos(2\theta) = 2\cos^2(\theta) - 1$$

or

$$\cos(2\theta) = 1 - 2\sin^2(\theta)$$

For tangent, the double angle identity is given by:

$$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$$

These identities allow for the simplification of expressions involving trigonometric functions. For example, if you encounter an expression like $$\cos^2\left(\frac{\pi}{12}\right) - \sin^2\left(\frac{\pi}{12}\right)$$, you can recognize it as the cosine double angle identity, simplifying it to $$\cos\left(\frac{\pi}{6}\right)$$, which equals $$\frac{\sqrt{3}}{2}$$.

When to use these identities often depends on the structure of the expression. If the argument of a function is double an angle, such as $$2\theta$$, it is a clear indication to apply the double angle identities. Additionally, recognizing parts of these identities within more complex expressions can guide you in simplification. For instance, if you have $$\sin(15^\circ) \cos(15^\circ)$$, you can identify it as part of the sine double angle identity. By manipulating the identity, you can rewrite it as:

$$\frac{\sin(30^\circ)}{2}$$

As you continue to learn and practice with these identities, your ability to simplify trigonometric expressions will improve, making it easier to tackle more complex problems. Mastery of double angle identities is a stepping stone to deeper understanding in trigonometry.

• A. 1
• B. $-\csc^2\theta$
• C. $\cot^2\theta$
• D. $-\cot^2\theta$
• E. $1+\sec\theta$
• F. $\frac{1}{\tan^2\theta}$
• G. $-\cot^2\theta\left(1+\sec\theta\right)$
• H. $-\tan^2\theta\left(1+\sec\theta\right)$
• I. $\frac{\sqrt2-\sqrt6}{4}$
• J. $\frac{\sqrt6-\sqrt2}{4}$
• K. $\frac{\sqrt2-\sqrt6}{2}$
• L. $\frac{\sqrt4}{4}$
• M. $\frac{\sqrt2-\sqrt6}{4}$
• N. $\frac{\sqrt6-\sqrt2}{4}$
• O. $\frac{\sqrt2-\sqrt6}{2}$
• P. $\frac{\sqrt4}{4}$
• Q. $\frac{\sqrt2-\sqrt6}{4}$
• R. $\frac{\sqrt6-\sqrt2}{4}$
• S. $\frac{\sqrt2-\sqrt6}{2}$
• T. $\frac{\sqrt4}{4}$
• U. $-\frac12$
• V. 0
• W. $\frac12$
• X. $\frac{\sqrt3}{2}$
• Y. $-\sin\theta-\cos\theta$
• Z. 0
• [. $-\sin\theta$
• \. $-\cos\theta$

Answer: (D)

Pythagorean Identities Example 2

In this problem, we aim to rewrite the expression \(\frac{1}{1 + \cos(\theta)}\) without a fraction by utilizing Pythagorean identities. Initially, the expression does not contain any squared trigonometric functions, which makes it less straightforward to apply these identities. However, we can manipulate the expression to achieve our goal.

To begin, we multiply both the numerator and the denominator by \(1 - \cos(\theta)\). This step is crucial as it maintains the value of the expression while allowing us to transform the denominator. The multiplication yields:

Numerator: \(1 \cdot (1 - \cos(\theta)) = 1 - \cos(\theta)\)

Denominator: \((1 + \cos(\theta))(1 - \cos(\theta)) = 1 - \cos^2(\theta)\)

Now, we recognize that \(1 - \cos^2(\theta)\) can be rewritten using the first Pythagorean identity, which states that:

\(1 = \sin^2(\theta) + \cos^2(\theta)\)

By rearranging this identity, we find that:

\(1 - \cos^2(\theta) = \sin^2(\theta)\)

Substituting this back into our expression, we have:

\(\frac{1 - \cos(\theta)}{\sin^2(\theta)}\)

To eliminate the fraction, we can express this as:

\(\frac{1}{\sin^2(\theta)}(1 - \cos(\theta))\)

Recognizing that \(\frac{1}{\sin(\theta)}\) is the cosecant function, we can rewrite \(\frac{1}{\sin^2(\theta)}\) as \(\csc^2(\theta)\). Thus, our expression simplifies to:

\(\csc^2(\theta)(1 - \cos(\theta))\)

In conclusion, we have successfully rewritten the original expression without a fraction, resulting in:

\(\frac{1}{1 + \cos(\theta)} = \csc^2(\theta)(1 - \cos(\theta))\)

Use the Pythagorean identities to rewrite the expression as a single term.
$\left(1+\csc\theta\right)\left(1-\csc\theta\right)$

• A. 1
• B. $-\csc^2\theta$
• C. $\cot^2\theta$
• D. $-\cot^2\theta$

Answer: (D)

Use the Pythagorean identities to rewrite the expression with no fraction.
$\frac{1}{1-\sec\theta}$

• A. $1+\sec\theta$
• B. $\frac{1}{\tan^2\theta}$
• C. $-\cot^2\theta\left(1+\sec\theta\right)$
• D. $-\tan^2\theta\left(1+\sec\theta\right)$

Answer: (C)

Sum and Difference of Sine & Cosine

In trigonometry, sum and difference identities are essential tools for simplifying expressions involving multiple angles. These identities allow us to break down complex trigonometric functions into simpler components, making it easier to evaluate them without a calculator, even for angles not directly on the unit circle.

For instance, consider the expression $\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\; +\; \backslash frac\{\backslash pi\}\{6\}\backslash right)$. To simplify this, we can apply the sum identity for sine, which states:

$\backslash sin(a\; +\; b)\; =\; \backslash sin(a)\; \backslash cos(b)\; +\; \backslash cos(a)\; \backslash sin(b)$

Here, $a\; =\; \backslash frac\{\backslash pi\}\{2\}$ and $b\; =\; \backslash frac\{\backslash pi\}\{6\}$. Substituting these values into the identity gives:

$\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)\; +\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; \backslash sin\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)$

From the unit circle, we know that $\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; 1$, $\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; 0$, $\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$, and $\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)\; =\; \backslash frac\{1\}\{2\}$. Thus, the expression simplifies to:

$1\; \backslash cdot\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}\; +\; 0\; \backslash cdot\; \backslash frac\{1\}\{2\}\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$

This demonstrates how the sum identity can simplify the evaluation of trigonometric functions. Similarly, the difference identity for sine is:

$\backslash sin(a\; -\; b)\; =\; \backslash sin(a)\; \backslash cos(b)\; -\; \backslash cos(a)\; \backslash sin(b)$

For cosine, the identities are slightly different. The sum identity is:

$\backslash cos(a\; +\; b)\; =\; \backslash cos(a)\; \backslash cos(b)\; -\; \backslash sin(a)\; \backslash sin(b)$

And the difference identity is:

$\backslash cos(a\; -\; b)\; =\; \backslash cos(a)\; \backslash cos(b)\; +\; \backslash sin(a)\; \backslash sin(b)$

These identities are particularly useful when dealing with angles that are not standard on the unit circle, such as 15 degrees. To find $\backslash cos(15^\backslash circ)$, we can express it as $\backslash cos(45^\backslash circ\; -\; 30^\backslash circ)$. Applying the cosine difference identity yields:

$\backslash cos(45^\backslash circ)\; \backslash cos(30^\backslash circ)\; +\; \backslash sin(45^\backslash circ)\; \backslash sin(30^\backslash circ)$

Substituting known values from the unit circle gives:

$\backslash frac\{\backslash sqrt\{2\}\}\{2\}\; \backslash cdot\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}\; +\; \backslash frac\{\backslash sqrt\{2\}\}\{2\}\; \backslash cdot\; \backslash frac\{1\}\{2\}\; =\; \backslash frac\{\backslash sqrt\{6\}\}\{4\}\; +\; \backslash frac\{\backslash sqrt\{2\}\}\{4\}\; =\; \backslash frac\{\backslash sqrt\{6\}\; +\; \backslash sqrt\{2\}\}\{4\}$

In summary, sum and difference identities are powerful tools in trigonometry that allow for the simplification and evaluation of trigonometric functions involving multiple angles. By recognizing when to apply these identities, particularly with angles that can be expressed as sums or differences of known values, students can effectively tackle a wide range of problems in trigonometry.

Sum and Difference of Sine & Cosine Example 3

In trigonometry, rewriting angles as the sum or difference of known angles on the unit circle can simplify the process of finding exact values for trigonometric functions. This technique is particularly useful for angles that are not commonly memorized, such as 75 degrees.

To illustrate this, consider the angle of 75 degrees. We can express it as the sum of two angles from the first quadrant of the unit circle. Notably, 30 degrees and 45 degrees can be combined since:

30° + 45° = 75°.

Using the sine sum formula, we can find:

sin(75°) = sin(30° + 45°) = sin(30°)cos(45°) + cos(30°)sin(45°).

Next, let's examine the cosine of -15 degrees. Here, we can use the angles 30 degrees and 45 degrees again. By subtracting, we find:

30° - 45° = -15°.

Thus, we can express this as:

cos(-15°) = cos(30° - 45°) = cos(30°)cos(45°) + sin(30°)sin(45°).

For angles in radians, such as 7π/12, we can also apply this method. The angles π/6, π/4, and π/3 are relevant here. To find a combination that sums to 7π/12, we can break it down as:

3π/12 + 4π/12 = 7π/12.

When simplified, this corresponds to:

π/4 + π/3.

Thus, we can express the cosine of 7π/12 as:

cos(7π/12) = cos(π/4 + π/3) = cos(π/4)cos(π/3) - sin(π/4)sin(π/3).

In summary, recognizing angles that are multiples of 15 degrees or π/12 radians allows us to effectively use sum and difference identities. This approach not only streamlines calculations but also enhances understanding of trigonometric relationships on the unit circle.

Find the exact value of the expression.
$\cos105\degree$

• A. $\frac{\sqrt2-\sqrt6}{4}$
• B. $\frac{\sqrt6-\sqrt2}{4}$
• C. $\frac{\sqrt2-\sqrt6}{2}$
• D. $\frac{\sqrt4}{4}$

Answer: (A)

Find the exact value of the expression.
$\sin15\degree$

• A. $\frac{\sqrt2-\sqrt6}{4}$
• B. $\frac{\sqrt6-\sqrt2}{4}$
• C. $\frac{\sqrt2-\sqrt6}{2}$
• D. $\frac{\sqrt4}{4}$

Answer: (B)

Find the exact value of the expression.
$\cos\frac{5\pi}{12}$

• A. $\frac{\sqrt2-\sqrt6}{4}$
• B. $\frac{\sqrt6-\sqrt2}{4}$
• C. $\frac{\sqrt2-\sqrt6}{2}$
• D. $\frac{\sqrt4}{4}$

Answer: (B)

Sum and Difference of Sine & Cosine Example 4

To find the exact value of the expression sin(10°) * cos(20°) + sin(20°) * cos(10°), we can utilize the sine sum formula. This expression resembles the sine addition formula, which states:

sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b)

In our case, we can identify a as 10 degrees and b as 20 degrees. Therefore, we can rewrite the expression as:

sin(10° + 20°)

Calculating the sum gives us:

sin(30°)

From the unit circle, we know that:

sin(30°) = 1/2

Thus, the exact value of the original expression is:

1/2

This approach highlights the importance of recognizing patterns in trigonometric identities, particularly the sine addition formula, to simplify complex expressions effectively.

Find the exact value of the expression.
$\cos80\degree\cos20\degree+\sin80\degree\sin20\degree$

• A. $-\frac12$
• B. 0
• C. $\frac12$
• D. $\frac{\sqrt3}{2}$

Answer: (C)

Sum and Difference of Sine & Cosine Example 5

In trigonometry, sum and difference identities are essential tools for simplifying expressions involving angles. These identities allow us to expand and simplify trigonometric functions that include variables. For instance, when expanding the expression for the sine of an angle plus a specific degree, we can apply the sum formula for sine.

Consider the expression sin(β + 30°). Using the sum formula, we can expand this to:

\[\sin(β + 30°) = \sin(β) \cos(30°) + \cos(β) \sin(30°)\]

From the unit circle, we know that cos(30°) = \frac{\sqrt{3}}{2} and sin(30°) = \frac{1}{2}. Substituting these values into the equation gives:

\[\sin(β + 30°) = \sin(β) \cdot \frac{\sqrt{3}}{2} + \cos(β) \cdot \frac{1}{2}\]

To simplify further, we can rearrange the terms to factor out the common denominator:

\[\sin(β + 30°) = \frac{1}{2} \left( \sqrt{3} \sin(β) + \cos(β) \right)\]

This expression is now simplified using the sum formula.

Next, let's explore the cosine of an angle minus a variable, such as cos(π/4 - θ). Here, we apply the difference formula for cosine:

\[\cos(π/4 - θ) = \cos(π/4) \cos(θ) + \sin(π/4) \sin(θ)\]

Again, from the unit circle, we find that both cos(π/4) and sin(π/4) equal \frac{\sqrt{2}}{2}. Substituting these values results in:

\[\cos(π/4 - θ) = \frac{\sqrt{2}}{2} \cos(θ) + \frac{\sqrt{2}}{2} \sin(θ)\]

Since both terms share the same coefficient, we can factor it out:

\[\cos(π/4 - θ) = \frac{\sqrt{2}}{2} \left( \cos(θ) + \sin(θ) \right)\]

This expression is now fully expanded and simplified using the difference formula.

By mastering these identities, you can effectively simplify a wide range of trigonometric expressions involving variables, enhancing your problem-solving skills in trigonometry.

Expand the expression using the sum & difference identities and simplify.
$\sin\left(-\theta-\frac{\pi}{2}\right)$

• A. $-\sin\theta-\cos\theta$
• B. 0
• C. $-\sin\theta$
• D. $-\cos\theta$

Answer: (D)

Double Angle Identities

Understanding trigonometric identities is essential for simplifying trigonometric expressions and solving problems efficiently. Among these identities, double angle identities are particularly useful. They are derived from the sum formulas for sine, cosine, and tangent, specifically when the same angle is used twice.

The double angle identity for sine states that:

$$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$$

For cosine, the double angle identity can be expressed in three forms:

$$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$$

Alternatively, using the Pythagorean identity, it can also be written as:

$$\cos(2\theta) = 2\cos^2(\theta) - 1$$

or

$$\cos(2\theta) = 1 - 2\sin^2(\theta)$$

For tangent, the double angle identity is given by:

$$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$$

These identities allow for the simplification of expressions involving trigonometric functions. For example, if you encounter an expression like $$\cos^2\left(\frac{\pi}{12}\right) - \sin^2\left(\frac{\pi}{12}\right)$$, you can recognize it as the cosine double angle identity, simplifying it to $$\cos\left(\frac{\pi}{6}\right)$$, which equals $$\frac{\sqrt{3}}{2}$$.

When to use these identities often depends on the structure of the expression. If the argument of a function is double an angle, such as $$2\theta$$, it is a clear indication to apply the double angle identities. Additionally, recognizing parts of these identities within more complex expressions can guide you in simplification. For instance, if you have $$\sin(15^\circ) \cos(15^\circ)$$, you can identify it as part of the sine double angle identity. By manipulating the identity, you can rewrite it as:

$$\frac{\sin(30^\circ)}{2}$$

As you continue to learn and practice with these identities, your ability to simplify trigonometric expressions will improve, making it easier to tackle more complex problems. Mastery of double angle identities is a stepping stone to deeper understanding in trigonometry.

Here’s what students ask on this topic:

What are the even and odd trigonometric functions, and how are they used?

Even trigonometric functions, like cosine ($\cos$) and secant ($\sec$), satisfy the property $f_{(}-x)=f_{(}x)$. This means their graphs are symmetric about the y-axis. Odd trigonometric functions, like sine ($\sin$), tangent ($\tan$), and cosecant ($\csc$), satisfy $f_{(}-x)=-f_{(}x)$, showing symmetry about the origin. These properties simplify calculations involving negative angles. For example, $\cos (-\theta )=\cos \left(\theta \right)$ and $\sin (-\theta )=-\sin \left(\theta \right)$. Recognizing these identities helps in solving equations and simplifying expressions efficiently.

What are the Pythagorean identities, and how are they derived?

The Pythagorean identities are derived from the Pythagorean Theorem applied to the unit circle. The primary identity is ${\sin }^2\theta +{\cos }^2\theta =1$. By dividing this equation by ${\cos }^2\theta$, we get ${\tan }^2\theta +1={\sec }^2\theta$. Similarly, dividing by ${\sin }^2\theta$ gives $1+{\cot }^2\theta ={\csc }^2\theta$. These identities are used to simplify expressions and solve equations involving squared trigonometric functions.

How do you use sum and difference identities to simplify trigonometric expressions?

Sum and difference identities are used when the argument of a trigonometric function involves the sum or difference of two angles. For example, the sine sum identity is $\sin (A+B)=\sin \left(A\right)\cos \left(B\right)+\cos \left(A\right)\sin \left(B\right)$. To simplify $\sin (75°)$, rewrite it as $\sin (45°+30°)$. Using the identity, this expands to $\sin (45°)\cos (30°)+\cos (45°)\sin (30°)$. Substituting known values, the result is $\frac{\sqrt{6}+\sqrt{2}}{4}$. These identities are especially useful for non-unit circle angles.

What are double angle identities, and when should they be used?

Double angle identities simplify expressions involving twice an angle. For sine, the identity is $\sin \left(2\theta \right)=2\sin \left(\theta \right)\cos \left(\theta \right)$. For cosine, it is $\cos \left(2\theta \right)={\cos }^2\theta -{\sin }^2\theta$. For tangent, it is $\tan \left(2\theta \right)=\frac{2\tan \left(\theta \right)}{1-{\tan }^2\theta }$. Use these identities when the argument involves $2\theta$ or when simplifying expressions like ${\cos }^2\theta -{\sin }^2\theta$. For example, ${\cos }^2\left(\frac{\pi }{12}\right)-{\sin }^2\left(\frac{\pi }{12}\right)$ simplifies to $\cos \left(\frac{\pi }{6}\right)$, which equals $\frac{\sqrt{3}}{2}$.

How do you verify a trigonometric identity?

To verify a trigonometric identity, start by simplifying one or both sides of the equation to show they are equal. Use known identities like Pythagorean, sum and difference, or double angle identities. For example, to verify ${\sin }^2\theta +{\cos }^2\theta =1$, recognize it as the Pythagorean identity. For more complex cases, rewrite terms using equivalent forms. For instance, to verify $\tan \left(\theta \right)=\frac{\sin }{\left(}$, rewrite $\tan \left(\theta \right)$ as $\frac{\sin }{\left(}$. Simplify step by step, ensuring each transformation is valid. Always aim to match the target expression exactly.