Cofunctions of Complementary Angles: Videos & Practice Problems

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, which can be expressed mathematically 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 \( \sin(30^\circ) \), calculate:

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

For angles expressed in radians, the same principle applies, but the complementary angle is calculated using \( \frac{\pi}{2} \). For example, if you need to find the cofunction for \( \cos\left(\frac{5\pi}{18}\right) \), you would compute:

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

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

\( \frac{\pi}{2} = \frac{9\pi}{18} \)

Then, subtract:

\( \sin\left(\frac{9\pi}{18} - \frac{5\pi}{18}\right) = \sin\left(\frac{4\pi}{18}\right) = \sin\left(\frac{2\pi}{9}\right) \)

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

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:

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

To solve for \(x\), we can rewrite the cosine term using the cofunction identity:

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

This transforms the equation into:

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

Now, we can set the arguments equal:

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

Solving this equation involves basic algebra. First, add \(x\) to both sides:

$$2x - 10 = 90^\circ$$

Next, add 10 to both sides:

$$2x = 100^\circ$$

Finally, divide by 2:

$$x = 50^\circ$$

This demonstrates how cofunction identities facilitate solving trigonometric equations.

For a more complex example, consider the equation:

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

Again, we can apply the cofunction identity:

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

This leads to:

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

Setting the arguments equal gives us:

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

To isolate \(\theta\), add \(\theta\) to both sides:

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

Next, add 30 to both sides:

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

Finally, divide by 3:

$$\theta = 40^\circ$$

This process illustrates the effectiveness of cofunction identities in solving for unknown angles in trigonometric equations. Mastering these identities will enhance your ability to tackle a variety of problems in trigonometry.