Double Angle Identities: Videos & Practice Problems

Double angle identities are derived from sum formulas for the same angle, enhancing the ability to simplify trigonometric expressions. Key identities include: sin(2θ)=2sin(θ)cos(θ), cos(2θ)=cos(θ)^2-sin(θ)^2, and tan(2θ)=2tan(θ)/(1-tan(θ)^2). Recognizing parts of these identities aids in simplification, allowing for efficient problem-solving in trigonometry.

Double Angle Identities

Trigonometric identities are essential tools in simplifying and solving trigonometric expressions. Among these, double angle identities are particularly useful, derived from the sum formulas for sine, cosine, and tangent when the same angle is used twice. Understanding these identities allows for more efficient problem-solving in trigonometry.

The double angle identity for sine states that:

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

This identity shows that the sine of double an angle can be expressed as twice the product of the sine and cosine of the angle.

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

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

These variations provide flexibility in simplifying expressions depending on the context.

The double angle identity for tangent is given by:

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

Recognizing when to apply these identities is crucial. If an expression involves terms like $\sin(2\theta)$ or $\cos(2\theta)$, it is a clear indication to use the corresponding double angle identity. Additionally, if part of an expression resembles a known identity, it can often be simplified using that identity.

For example, consider the expression $\cos^2\left(\frac{\pi}{12}\right) - \sin^2\left(\frac{\pi}{12}\right)$. This can be recognized as the cosine double angle identity, allowing us to rewrite it as:

$$\cos\left(2 \cdot \frac{\pi}{12}\right) = \cos\left(\frac{\pi}{6}\right)$$

Knowing that $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$ simplifies the expression significantly.

Another example involves simplifying the expression $\sin(15^\circ) \cos(15^\circ)$. By recognizing this as part of the sine double angle identity, we can rewrite it as:

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

Thus, the expression simplifies to $\frac{1}{2}$, since $\sin(30^\circ) = \frac{1}{2}$.

As you continue to learn and practice with trigonometric identities, the ability to recognize and apply these double angle identities will enhance your problem-solving skills and make tackling trigonometric expressions more manageable.

Double Angle Identities Example 1

In this problem, we are tasked with verifying the identity that states the sine of \(2\theta\) over the tangent of \(\theta\) is equal to the cosine of \(2\theta\) plus 1. To approach this verification, we start with the more complex side, which in this case is the left side containing a fraction.

We begin by applying the double angle formula for sine, which states that:

\[\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\]

Thus, we can rewrite the left side as:

\[\frac{\sin(2\theta)}{\tan(\theta)} = \frac{2 \sin(\theta) \cos(\theta)}{\tan(\theta)}\]

Next, we express the tangent function in terms of sine and cosine:

\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]

Substituting this into our expression gives us:

\[\frac{2 \sin(\theta) \cos(\theta)}{\frac{\sin(\theta)}{\cos(\theta)}}\]

To simplify this fraction, we multiply by the reciprocal of the denominator:

\[2 \sin(\theta) \cos(\theta) \cdot \frac{\cos(\theta)}{\sin(\theta)} = 2 \cos^2(\theta)\]

Now, we need to relate \(2 \cos^2(\theta)\) to the right side of the identity, which is \(\cos(2\theta) + 1\). We recall the cosine double angle formula:

\[\cos(2\theta) = 2 \cos^2(\theta) - 1\]

By rearranging this formula, we can express \(2 \cos^2(\theta)\) as:

\[2 \cos^2(\theta) = \cos(2\theta) + 1\]

Thus, we can substitute back into our expression:

\[2 \cos^2(\theta) = \cos(2\theta) + 1\]

Now we see that the left side simplifies to \(\cos(2\theta) + 1\), which matches the right side of the original identity. Therefore, we have successfully verified the identity:

\[\frac{\sin(2\theta)}{\tan(\theta)} = \cos(2\theta) + 1\]

In conclusion, through the application of double angle formulas and simplification techniques, we have confirmed the identity is valid. If you have any questions or need further clarification, feel free to ask!

Double Angle Identities Example 2

In this problem, we are tasked with verifying the identity involving the cotangent function. Specifically, we need to show that the cotangent of \(2\theta\) is equal to the expression \(\cot(\theta) - \frac{1}{2} \cos(\theta) \sin(\theta)\). To approach this, we start with the more complex side of the equation, which is the right side.

First, we recall that the cotangent function can be expressed in terms of sine and cosine: \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\). Substituting this into our expression gives us:

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

Next, we need to combine these fractions. The common denominator for the two fractions is \(2 \cos(\theta) \sin(\theta)\). We rewrite the first term to have this common denominator:

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

Now, we can recognize a trigonometric identity in the numerator. The expression \(2 \cos^2(\theta) - 1\) can be rewritten using the double angle identity for cosine, which states that \(\cos(2\theta) = 2 \cos^2(\theta) - 1\). Thus, we can express the numerator as:

\[\cos(2\theta)\]

For the denominator, we can use the double angle identity for sine, which states that \(\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\). Therefore, we can rewrite the denominator as:

\[\sin(2\theta)\]

Putting it all together, we have:

\[\frac{\cos(2\theta)}{\sin(2\theta)}\]

This expression is equivalent to \(\cot(2\theta)\), which is exactly what we set out to verify. Thus, we have successfully shown that:

\[\cot(2\theta) = \cot(\theta) - \frac{1}{2} \cos(\theta) \sin(\theta)\]

In conclusion, both sides of the identity are equal, confirming the validity of the original statement.

Double Angle Identities Example 3

In trigonometry, when tasked with finding a function of double angles, such as the sine of \(2\theta\), it is essential to utilize double angle identities effectively. The identity for the sine of double angles is given by:

\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]

To solve for \(\sin(2\theta)\) when \(\cos(\theta) = \frac{4}{5}\) and \(\theta\) is in the first quadrant (where both sine and cosine are positive), we can follow a systematic approach.

First, we need to identify the unknown trigonometric values. Since we know \(\cos(\theta)\), we can visualize this using a right triangle. In this triangle, the adjacent side is 4 and the hypotenuse is 5. To find the opposite side, we can apply the Pythagorean theorem:

\[ a^2 + b^2 = c^2 \]

Here, \(a\) is the opposite side, \(b\) is the adjacent side (4), and \(c\) is the hypotenuse (5). Thus:

\[ a^2 + 4^2 = 5^2 \]

\[ a^2 + 16 = 25 \]

\[ a^2 = 9 \]

\[ a = 3 \]

Now, we can determine \(\sin(\theta)\) as follows:

\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} \]

With both \(\sin(\theta)\) and \(\cos(\theta)\) known, we can substitute these values into the double angle identity:

\[ \sin(2\theta) = 2 \cdot \sin(\theta) \cdot \cos(\theta) = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} \]

Calculating this gives:

\[ \sin(2\theta) = 2 \cdot \frac{12}{25} = \frac{24}{25} \]

Thus, the final result for \(\sin(2\theta)\) is:

\[ \sin(2\theta) = \frac{24}{25} \]

This methodical approach not only reinforces the understanding of trigonometric identities but also highlights the importance of visualizing problems through right triangles and applying the Pythagorean theorem to find missing values.

Given $\tan\theta=\frac{5}{12}$ and 0 < $\theta$ < $\frac{\pi}{2}$, find $\cos\left(2\theta\right)$.

• A. 0
• B. $-\frac{199}{169}$
• C. $\frac{119}{169}$
• D. $\frac{144}{169}$

Answer: (C)

Here’s what students ask on this topic:

What are the double angle identities in trigonometry?

Double angle identities are trigonometric formulas that express trigonometric functions of double angles (2θ) in terms of single angles (θ). The key double angle identities are:

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

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

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

How do you derive the double angle identity for sine?

The double angle identity for sine is derived from the sum formula for sine. The sum formula states:

$\sin (A+B)=\sin \left(A\right)\cos \left(B\right)+\cos \left(A\right)\sin \left(B\right)$

For the same angle θ, we have:

$\sin (\theta +\theta )=\sin \left(\theta \right)\cos \left(\theta \right)+\cos \left(\theta \right)\sin \left(\theta \right)$

Since $\sin \left(\theta \right)\cos \left(\theta \right)=\cos \left(\theta \right)\sin \left(\theta \right)$, this simplifies to:

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

When should you use double angle identities in trigonometry?

Double angle identities should be used whenever the argument of a trigonometric function is twice an angle, i.e., 2θ. They are particularly useful for simplifying expressions and solving equations involving trigonometric functions. Additionally, if you recognize parts of the double angle identities within an expression, you can use these identities to simplify the expression further. For example, if you see $2\sin \left(\theta \right)\cos \left(\theta \right)$, you can replace it with $\sin \left(2\theta \right)$. Recognizing these patterns helps in solving trigonometric problems more efficiently.

How do you simplify trigonometric expressions using double angle identities?

To simplify trigonometric expressions using double angle identities, follow these steps:

1. Identify parts of the expression that match double angle identities. For example, look for terms like $2\sin \left(\theta \right)\cos \left(\theta \right)$ or $\cos \left(\theta \right)^2-\sin \left(\theta \right)^2$.
2. Replace these parts with their corresponding double angle identities. For example, $2\sin \left(\theta \right)\cos \left(\theta \right)$ becomes $\sin \left(2\theta \right)$.
3. Simplify the resulting expression further if possible.

For example, to simplify $\sin (15°)\cos (15°)$, recognize it as part of the sine double angle identity and rewrite it as $\frac{\sin }{(}$.

What are the alternate forms of the cosine double angle identity?

The cosine double angle identity has three forms, derived from the Pythagorean identities. The primary form is:

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

Using the Pythagorean identity $\sin \left(\theta \right)^2=1-\cos \left(\theta \right)^2$, we get:

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

Similarly, using $\cos \left(\theta \right)^2=1-\sin \left(\theta \right)^2$, we get:

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

These alternate forms are useful in different scenarios for simplifying trigonometric expressions.