Cofunctions of Complementary Angles: Videos & Practice Problems

Understanding cofunction identities and complementary angles is essential in trigonometry. Complementary angles add up to 90 degrees, and cofunctions like sine and cosine are equal for these angles. For example, $\sin \left(\mathrm{53}\right)=\cos \left(\mathrm{37}\right)$. This relationship aids in solving equations by allowing the rewriting of one side to match the other, facilitating the setting of arguments equal for straightforward solutions.

Cofunction Identities

In the study of trigonometric functions, understanding cofunction identities and complementary angles is essential. Complementary angles are defined as two angles that sum to 90 degrees. In the context of a right triangle, the two non-right angles are always complementary, as one angle is fixed at 90 degrees, making the sum of the other two angles equal to 90 degrees.

The complementary angle theorem states that the cofunctions of complementary angles are equal. This means that for any angle \( \theta \), the sine of that angle is equal to the cosine of its complementary angle. Mathematically, this can be expressed as:

\[ \sin(\theta) = \cos(90^\circ - \theta) \]

For example, if \( \theta = 53^\circ \), then:

\[ \sin(53^\circ) = \cos(37^\circ) \]

Similarly, other pairs of cofunctions include:

• Secant and cosecant: \( \sec(\theta) = \csc(90^\circ - \theta) \)
• Tangent and cotangent: \( \tan(\theta) = \cot(90^\circ - \theta) \)

To find the cofunction for a given angle, subtract the angle from 90 degrees. For instance, to find the cofunction for \( \sin(30^\circ) \), we calculate:

\[ \cos(90^\circ - 30^\circ) = \cos(60^\circ) \]

For \( \tan(16^\circ) \), the cofunction is:

\[ \cot(90^\circ - 16^\circ) = \cot(74^\circ) \]

When dealing with angles in radians, the same principle applies, but we use \( \frac{\pi}{2} \) instead of 90 degrees. For example, to find the cofunction for \( \cos\left(\frac{5\pi}{18}\right) \), we first recognize that the cofunction is sine:

\[ \sin\left(\frac{\pi}{2} - \frac{5\pi}{18}\right) \]

To simplify this, convert \( \frac{\pi}{2} \) to a fraction with a common denominator:

\[ \frac{9\pi}{18} - \frac{5\pi}{18} = \frac{4\pi}{18} = \frac{2\pi}{9} \]

Thus, \( \cos\left(\frac{5\pi}{18}\right) = \sin\left(\frac{2\pi}{9}\right) \).

Understanding these relationships between trigonometric functions and their cofunctions is crucial for solving various problems in trigonometry, especially those involving right triangles and angle measures.

Write the expression in terms of the appropriate cofunction.
$\cos\left(\frac{19\pi}{45}\right)$

• A. $\cos\left(\frac{7\pi}{90}\right)$
• B. $\sin\left(\frac{4031\pi}{45}\right)$
• C. $\sin\left(\frac{7\pi}{90}\right)$
• D. $\cos\left(\frac{4031\pi}{90}\right)$

Answer: (C)

Write the expression in terms of the appropriate cofunction.
$\cot\left(25\degree\right)$

• A. $\tan\left(65\degree\right)$
• B. $\tan\left(25\degree\right)$
• C. $\cot\left(65\degree\right)$
• D. $\cot\left(25\degree\right)$

Answer: (A)

Using Cofunction Identities to Solve Equations

Understanding cofunction identities is essential for solving trigonometric equations, as these identities reveal the relationships between certain trigonometric functions and their complementary angles. For instance, the sine and cosine functions are cofunctions, meaning that:

$$\sin(x) = \cos(90^\circ - x)$$

When faced with an equation involving different trigonometric functions, the first step is to rewrite one side of the equation using cofunction identities to create a common function on both sides. This allows for setting the arguments of the functions equal to each other, simplifying the solving process.

For example, consider the equation:

$$\sin(x) - 10 = \cos(x)$$

By applying the cofunction identity, we can rewrite the cosine term:

$$\sin(x) - 10 = \sin(90^\circ - x)$$

This leads to:

$$x - 10 = 90^\circ - x$$

Now, solving for \(x\) becomes straightforward. By adding \(x\) to both sides and then adding 10 to both sides, we find:

$$2x = 100^\circ$$

Dividing both sides by 2 gives:

$$x = 50^\circ$$

In a more complex example, consider the equation:

$$\cos(\theta) = \sin(2\theta - 30^\circ)$$

Again, we can use the cofunction identity to rewrite the cosine term:

$$\sin(90^\circ - \theta) = \sin(2\theta - 30^\circ)$$

Setting the insides equal gives:

$$90^\circ - \theta = 2\theta - 30^\circ$$

To solve for \(\theta\), we first add \(\theta\) to both sides:

$$90^\circ = 3\theta - 30^\circ$$

Next, we add 30 to both sides:

$$120^\circ = 3\theta$$

Finally, dividing by 3 yields:

$$\theta = 40^\circ$$

Through these examples, it is clear that cofunction identities are powerful tools for solving trigonometric equations, allowing for the manipulation of functions to find unknown angles effectively.

Find the acute angle solution to the following equation involving cofunctions. $\theta$ is in degrees.
$\cos\left(2\theta+15\right)=\sin\left(5\theta+12\right)$

• A. $1\degree$
• B. $4\degree$
• C. $6\degree$
• D. $9\degree$

Answer: (D)

Find the acute angle solution to the following equation involving cofunctions. P is in degrees.
$\sec\left(\frac{4P}{5}+20\right)=\csc\left(\frac{5P}{8}+\frac{223}{4}\right)$

• A. $5\degree$
• B. $10\degree$
• C. $12\degree$
• D. $15\degree$

Answer: (B)

Find the acute angle solution to the following equation involving cofunctions. M is in radians.
$\tan\left(\frac{M}{2}+5\right)=\cot\left(M-5\right)$

• A. $\frac{\pi}{2}$
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{3}$
• D. $\frac{\pi}{6}$

Answer: (C)

Here’s what students ask on this topic:

What are cofunction identities in trigonometry?

Cofunction identities in trigonometry describe the relationship between pairs of trigonometric functions whose angles are complementary, meaning they add up to 90 degrees. For example, the sine and cosine functions are cofunctions because $\sin \left(A\right)=\cos (90°-A)$. Similarly, the tangent and cotangent, secant and cosecant functions are also cofunctions. These identities are useful for solving trigonometric equations and understanding the properties of right triangles.

How do you find the cofunction of an angle?

To find the cofunction of an angle, you need to identify the complementary angle, which is $90°-A$ for an angle $A$. For example, if you have $\sin (30°)$, the cofunction is $\cos (90°-30°)=\cos (60°)$. This principle applies to all trigonometric functions and their cofunctions, such as tangent and cotangent, secant and cosecant.

Why are cofunction identities important in solving trigonometric equations?

Cofunction identities are important in solving trigonometric equations because they allow you to rewrite one side of an equation to match the other side, making it easier to solve. For example, if you have an equation like $\sin \left(x\right)=\cos \left(y\right)$, you can use the cofunction identity $\cos \left(y\right)=\sin (90°-y)$ to rewrite the equation as $\sin \left(x\right)=\sin (90°-y)$. This allows you to set the arguments equal and solve for the unknown variable.

How do you use cofunction identities to solve trigonometric equations?

To use cofunction identities to solve trigonometric equations, follow these steps: 1) Identify the cofunction identity that relates the trigonometric functions in your equation. For example, $\sin \left(A\right)=\cos (90°-A)$. 2) Rewrite one side of the equation using the cofunction identity. 3) Set the arguments of the trigonometric functions equal to each other. 4) Solve for the unknown variable using algebraic methods. For example, if you have $\sin \left(x\right)=\cos \left(y\right)$, rewrite it as $\sin \left(x\right)=\sin (90°-y)$, then set $x=90°-y$ and solve for $x$ or $y$.

What is the complementary angle theorem?

The complementary angle theorem states that the cofunctions of complementary angles are equal. In other words, if two angles add up to 90 degrees, the sine of one angle is equal to the cosine of the other angle, and vice versa. Mathematically, this can be expressed as $\sin \left(A\right)=\cos (90°-A)$ and $\cos \left(A\right)=\sin (90°-A)$. This theorem is useful for solving trigonometric equations and understanding the properties of right triangles.