Cofunctions of Complementary Angles and Cofunction Identities

Cofunctions of Complementary Angles

Definition of Complementary Angles

Complementary angles are two angles whose measures add up to 90°. In a right triangle, the two non-right angles are always complementary.

• Complementary Angle Theorem: Cofunctions of complementary angles are equal.

Cofunction Identities

Cofunction identities relate the trigonometric functions of complementary angles. For any angle $A$:

• $\sin A = \cos(90^{\circ} - A)$

• $\cos A = \sin(90^{\circ} - A)$

• $\tan A = \cot(90^{\circ} - A)$

• $\cot A = \tan(90^{\circ} - A)$

• $\sec A = \csc(90^{\circ} - A)$

• $\csc A = \sec(90^{\circ} - A)$

Example: If $\sin 37^{\circ} = x$, then $\cos 53^{\circ} = x$ because $37^{\circ} + 53^{\circ} = 90^{\circ}$.

Using Cofunction Identities

To write a trigonometric expression in terms of its cofunction, use the identities above. For example:

• $\sin 30^{\circ} = \cos 60^{\circ}$

• $\tan 54^{\circ} = \cot 36^{\circ}$

• $\sec 10^{\circ} = \csc 80^{\circ}$

• $\csc 18^{\circ} = \sec 72^{\circ}$

Practice Problems

• Write $\tan(48^{\circ})$ in terms of its cofunction: $\cot(42^{\circ})$

• Write $\cos(25^{\circ})$ in terms of its cofunction: $\sin(65^{\circ})$

Solving Equations Using Cofunction Identities

Solving Trigonometric Equations

To solve equations involving cofunction identities, rewrite one side using the appropriate identity so both sides have the same function, then set the arguments equal.

• Example: $\sin(x) = \cos(90^{\circ} - x)$

• Rewrite: $\sin(x) = \sin(x)$, so $x = x$

For more complex equations:

• Example: $\cos(2x + 15) = \sin(34 + x)$

• Rewrite $\sin(34 + x)$ as $\cos(56 - x)$

• Set arguments equal: $2x + 15 = 56 - x$

• Solve: $3x = 41 \implies x = 13.67$

How To: Steps for Solving Using Cofunctions

1. Use cofunction identities to get the same trig function on both sides.

2. Set the insides (arguments) of the functions equal.

3. Solve for the missing variable.

Practice Problems

• Find the acute angle solution $P$ in degrees: $\cos(2P + 15) = \sin(34 + 17)$ Solution: $2P + 15 = 56 - 17 \implies 2P + 15 = 39 \implies 2P = 24 \implies P = 12$

• Find the acute angle solution $P$ in degrees: $\cos\left(\frac{P}{2} + 20\right) = \sin\left(70 - \frac{P}{2}\right)$ Solution: $\frac{P}{2} + 20 = 20 + \frac{P}{2}$ (after using cofunction identity and simplifying)

• Find the acute angle solution $x$ in radians: $\tan\left(\frac{\pi}{2} - x\right) = \cos(x - 1)$ Solution: $\cot(x) = \cos(x - 1)$ (requires further steps)

Additional info: Cofunction identities are essential for simplifying trigonometric expressions and solving equations, especially in right triangle trigonometry and when working with complementary angles.