Precalculus by Richard Wright

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5-05 Sum and Difference Formulas

Mr. Wright teaches the lesson.

Summary: In this section, you will:

Apply the sum and difference formulas to evaluate trigonometric expressions.
Apply the sum and difference formulas to simplify trigonometric expressions.
Apply the sum and difference formulas to solve trigonometric equations.

SDA NAD Content Standards (2018): PC.5.1

The Matterhorn in the Swiss Alps. (pixabay/Christophe Schindler)

How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities were often derived from real-world applications, including their use in calculating long distances.

The trigonometric identities in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that the terms formula and identity are often used interchangeably.

Sum and Difference Formulas

The proofs for the sum and difference formulas can be found at Wikipedia.

Sum and Difference Formulas

sin(u + v) = sin u cos v + cos u sin v
sin(u − v) = sin u cos v − cos u sin v
cos(u + v) = cos u cos v − sin u sin v
cos(u − v) = cos u cos v + sin u sin v
$\tan\left(u + v\right) = \frac{\tan u + \tan v}{1 - \tan u \tan v}$
$\tan\left(u - v\right) = \frac{\tan u - \tan v}{1 + \tan u \tan v}$

These identities can be used to evaluate trigonometric expressions, simplify trigonometric expressions, derive new identities, and solve trigonometric equations.

Evaluate a Trigonometric Expression

Use a sum or difference formula to find the exact value of cos 165°.

Solution

165° is not a special angle on the unit circle, but 165° = 135° + 30°, which are both special angles on the unit circle. So the sum formula for cosine can be used where u = 135° and v = 30°.

cos 165° = cos(135° + 30°)

= cos 135° cos 30° − sin 135° sin 30°

$$ = \left(-\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)$$

$$ =\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$$

$$ =\frac{\sqrt{6}-\sqrt{2}}{4}$$

Find the exact value of $\tan \frac{7π}{12}$.

Answer

$-\sqrt{3} - 2$

Evaluate a Trigonometric Expression

Find the exact value of sin 200° cos 50° − cos 200° sin 50°.

Solution

This looks a lot like the right-hand side of the sine difference formula. Use it with u = 200° and v = 50°.

sin 200° cos 50° − cos 200° sin 50°

= sin(200° − 50°)

= sin 150°

$$=\frac{1}{2}$$

Find the exact value of cos 100° cos 35° − sin 100° sin 35°.

Answer

$-\frac{\sqrt{2}}{2}$

Derive a New Identity

Derive a formula for cos(t + π).

Solution

Because angles are added in the cosine function, use the cosine sum formula with u = t and v = π.

cos(t + π)

cos t cos π − sin t sin π

cos t (−1) − sin t (0)

−cos t

Derive a formula for tan(x + π).

Answer

tan x

Verify an Identity

Verify $\tan x \sin\left(x + \frac{π}{2}\right) = \sin x$

Solution

The left side is more complicated, so start there. Use the sine sum formula with u = x and $v = \frac{π}{2}$ to simplify the sine expression.

$$\tan x \sin\left(x + \frac{π}{2}\right) = \sin x$$

$$\tan x \left(\sin x \cos \frac{π}{2} + \cos x \sin \frac{π}{2} \right) = \sin x$$

tan x (sin x (0) + cos x (1)) = sin x

tan x (cos x) = sin x

Rewrite tan x as $\frac{\sin x}{\cos x}$.

$$\frac{\sin x}{\cos x} \cos x = \sin x$$

The cosines cancel.

sin x = sin x

Verify $\cos\left(x - \frac{π}{2}\right) = \sin x$.

Answer

$\cos\left(x - \frac{π}{2}\right)$ $= \cos x \cos \frac{π}{2} + \sin x \sin \frac{π}{2}$ $= \left(\cos x\right)\left(0\right) + \left(\sin x\right)\left(1\right)$ $= \sin x$

Solve a Trigonometric Equation

Find all the solutions in the interval [0, 2π) of $\sin\left(x + \frac{2π}{3}\right) + \sin\left(x - \frac{2π}{3}\right) = 1$.

Solution

There are two places to use sum or difference formulas. In both u = x and $v = \frac{2π}{3}$.

$$\left(\sin x \cos \frac{2π}{3} + \cos x \sin \frac{2π}{3}\right) + \left(\sin x \cos \frac{2π}{3} - \cos x \sin \frac{2π}{3}\right) = 1$$

The $\cos x \sin \frac{2π}{3}$ terms cancel out, and the $\sin x \cos \frac{2π}{3}$ terms add together.

$$2\sin x \cos \frac{2π}{3} = 1$$

$$2\sin x \left(-\frac{1}{2}\right) = 1$$

−sin x = 1

sin x = −1

$$x = \frac{3π}{2}$$

Find all the solutions in the interval [0, 2π) of sin(x + π) + sin(x − π) = 1.

Answer

$\frac{7π}{6}$, $\frac{11π}{6}$

Lesson Summary

Sum and Difference Formulas

sin(u + v) = sin u cos v + cos u sin v
sin(u − v) = sin u cos v − cos u sin v
cos(u + v) = cos u cos v − sin u sin v
cos(u − v) = cos u cos v + sin u sin v
$\tan\left(u + v\right) = \frac{\tan u + \tan v}{1 - \tan u \tan v}$
$\tan\left(u - v\right) = \frac{\tan u - \tan v}{1 + \tan u \tan v}$

Helpful videos about this lesson.

Mr. Wright Teaches the Lesson (https://youtu.be/aJg4DfHUOBM)
Sum and Difference Identities for Cosine (http://openstaxcollege.org/l/sumdifcos)
Sum and Difference Identities for Sine (http://openstaxcollege.org/l/sumdifsin)
Sum and Difference Identities for Tangent (http://openstaxcollege.org/l/sumdiftan)

Practice Exercises

Use a sum or difference formula to find the exact value of the expression.
sin 255°
$\tan \frac{5π}{12}$
cos 230° cos 20° + sin 230° sin 20°
$\cos \frac{11π}{12}$
Derive a formula for the following expressions.
tan(x − π)
$\sin \left(\frac{3π}{2} - x\right)$
Verify the following identities.
sec x sin(π + x) = −tan x
$\tan\left(π + x\right) \sin \left(\frac{3π}{2} - x\right) = -\sin x$
$\frac{\cos\left(x + h\right) - \cos x}{h} = \cos x \left(\frac{\cos h - 1}{h}\right) - \sin x\left(\frac{\sin h}{h}\right)$
Solve the trigonometric equations on the interval [0, 2π).
sin(2x − 3π) = 0
$\cos\left(x + \frac{3π}{4}\right) - \cos\left(x - \frac{3π}{4}\right) = 1$
sin(π + x) − sin(π − x) = 2
Mixed Review
(5-04) Find all the solutions of 2 tan x + 1 = 3.
(5-04) Use a graphing utility to find the solutions in the interval [0, 2π) to three decimal places: sin x + cos x = 1.
(5-03) Verify the identity graphically: $\frac{\cos x - \cos^3 x}{\sin x} = \cos x \sin x$.

Answers

$\frac{-\sqrt{6}-\sqrt{2}}{4}$
$\sqrt{3} + 2$
$-\frac{\sqrt{3}}{2}$
$\frac{-\sqrt{6}-\sqrt{2}}{4}$
tan x
−cos x
$\sec x \sin \left(π + x\right)$ $= \sec x\left(\sin π \cos x + \cos π \sin x\right)$ $= \sec x \left(-\sin x\right)$ $= -\tan x$
$\tan\left(π + x\right) \sin \left(\frac{3π}{2} - x\right)$ $= \left(\frac{\tan π + \tan x}{1 - \tan π \tan x}\right)\left(\sin \frac{3π}{2} \cos x - \cos \frac{3π}{2} \sin x\right)$ $= (\tan x)(-\cos x)$ $= -\sin x$
$\frac{\cos\left(x + h\right) - \cos x}{h}$ $= \frac{\cos x \cos h - \sin x \sin h - \cos x}{h}$ $= \frac{\cos x\left(\cos h - 1\right) - \sin x \sin h}{h}$ $= \cos x\left(\frac{\cos h - 1}{h}\right) - \sin x\left(\frac{\sin h}{h}\right)$
0, $\frac{π}{2}$, π, $\frac{3π}{2}$
$\frac{5π}{4}, \frac{7π}{4}$
$\frac{3π}{2}$
$\frac{π}{4} + πn$
0, $\frac{π}{2}$