Lesson Objectives
  • Learn how to use the difference identity for Cosine
  • Learn how to use the sum identity for Cosine

What are the Sum & Difference Identities for Cosine


In this lesson, we will cover the sum and difference identities for cosine.

Cosine of a Sum or Difference

Sum Identity for Cosine: $$\text{cos}(A + B)=\text{cos}A \hspace{.15em}\text{cos}B - \text{sin}A \hspace{.15em}\text{sin}B$$ Difference Identity for Cosine: $$\text{cos}(A - B)=\text{cos}A \hspace{.15em}\text{cos}B + \text{sin}A \hspace{.15em}\text{sin}B$$ There are many uses for these identities, let's start with some examples of how to find an exact function value.
Example #1: Find the exact value. $$\text{cos}\hspace{.1em}75°$$ To solve this problem, we need to realize that we are not able to just punch this into a calculator since we want the exact value and not an approximation. To solve this problem, we will break 75° up using special angles. $$\text{cos}\hspace{.1em}75°$$ Use our sum identity for cosine: $$\text{cos}(A + B)=\text{cos}A \hspace{.15em}\text{cos}B - \text{sin}A \hspace{.15em}\text{sin}B$$ $$\text{cos}\hspace{.1em}75°$$ $$=\text{cos}\hspace{.1em}(45° + 30°)$$ $$=\text{cos}\hspace{.1em}45° \hspace{.1em}\text{cos}\hspace{.1em}30° - \text{sin}\hspace{.1em}45° \hspace{.1em}\text{sin}\hspace{.1em}30°$$ $$=\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}- \frac{\sqrt{2}}{2}\cdot \frac{1}{2}$$ $$=\frac{\sqrt{6}}{4}- \frac{\sqrt{2}}{4}$$ $$=\frac{\sqrt{6}- \sqrt{2}}{4}$$ Example #2: Find the exact value. $$\text{cos}\hspace{.1em}\frac{π}{12}$$ Use our difference identity for cosine: $$\text{cos}(A - B)=\text{cos}A \hspace{.15em}\text{cos}B + \text{sin}A \hspace{.15em}\text{sin}B$$ $$\text{cos}\hspace{.1em}\frac{π}{12}$$ When given a problem with radians, it may be helpful to convert to degrees and then back to radians. It is easier to mentally work with degrees. $$\frac{π}{12}\cdot \frac{180°}{π}=15°$$ We know that we can obtain 15 degrees as the difference of 45° and 30°. Now, we can return to using radians. $$=\text{cos}\hspace{.1em}\left(\frac{π}{4}- \frac{π}{6}\right)$$ $$=\text{cos}\hspace{.1em}\frac{π}{4}\cdot \text{cos}\frac{π}{6}+ \text{sin}\frac{π}{4}\cdot \text{sin}\frac{π}{6}$$ $$=\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+ \frac{\sqrt{2}}{2}\cdot \frac{1}{2}$$ $$=\frac{\sqrt{6}}{4}+ \frac{\sqrt{2}}{4}$$ $$=\frac{\sqrt{6}+ \sqrt{2}}{4}$$ Example #3: Find the exact value. $$\text{cos}\hspace{.1em}83° \hspace{.1em}\text{cos}38° + \text{sin}\hspace{.1em}83° \hspace{.1em}\text{sin}38°$$ Use our difference identity for cosine: $$\text{cos}\hspace{.1em}83° \hspace{.1em}\text{cos}38° + \text{sin}\hspace{.1em}83° \hspace{.1em}\text{sin}38°$$ $$=\text{cos}(83° - 38°)$$ $$=\text{cos}\hspace{.1em}45°$$ $$=\frac{\sqrt{2}}{2}$$

Verifying Identities

We may also be asked to verify identities using our sum/difference identities. Let's look at some examples.
Example #4: Verify each identity. $$\text{cos}(θ + π)=-\text{cos}θ$$ Since the left side is more complex, let's flip the sides and work on the right: $$-\text{cos}θ=\text{cos}(θ + π)$$ $$=\text{cos}(θ + π)$$ $$=\text{cos}\hspace{.1em}θ \hspace{.1em}\text{cos}π - \text{sin}\hspace{.1em}θ \hspace{.1em}\text{sin}π$$ $$\text{cos}\hspace{.1em}π=-1$$ $$\text{sin}\hspace{.1em}π=0$$ Let's replace: $$=\text{cos}\hspace{.1em}θ \hspace{.1em}\text{cos}π - \text{sin}\hspace{.1em}θ \hspace{.1em}\text{sin}π$$ $$=\text{cos}\hspace{.1em}θ \hspace{.1em}\cdot -1 - \text{sin}\hspace{.1em}θ \hspace{.1em}\cdot 0$$ $$=\text{cos}\hspace{.1em}θ \hspace{.1em}\cdot -1$$ $$=-\text{cos}\hspace{.1em}θ \hspace{.1em}✓$$ Example #5: Verify each identity. $$\text{cos}^2 x - \text{sin}^2 x=\text{cos}\hspace{.1em}2x$$ In this case, we will work on the right side: $$\text{cos}^2 x - \text{sin}^2 x=\text{cos}\hspace{.1em}2x$$ $$=\text{cos}\hspace{.1em}2x$$ $$=\text{cos}(x + x)$$ $$=\text{cos}\hspace{.1em}x \cdot \text{cos}\hspace{.1em}x - \text{sin}\hspace{.1em}x \cdot \text{sin}\hspace{.1em}x$$ $$=\text{cos}^2 x - \text{sin}^2 x ✓$$

Finding cos(s + t) Given Information about s and t

We may be asked to find cos (s + t) when given some information about s and t. Let's look at an example.
Example #6: Find cos(s + t). $$\text{sin}\hspace{.1em}s=\frac{3}{5}$$ $$\text{sin}\hspace{.1em}t=-\frac{12}{13}$$ s in QI and t in QIII:
Let's start by finding the cosine of s. To do this we use the Pythagorean Identity: $$\text{cos}^2 s + \text{sin}^2 s=1$$ Plug in for sin s: $$\text{cos}^2 s + \left(\frac{3}{5}\right)^2=1$$ $$\text{cos}^2 s + \frac{9}{25}=1$$ $$\text{cos}^2 s=1 - \frac{9}{25}$$ $$\text{cos}^2 s=\frac{25}{25}- \frac{9}{25}$$ $$\text{cos}^2 s=\frac{16}{25}$$ Since s is in quadrant I, cosine is positive. Let's use the principal square root: $$\text{cos}\hspace{.1em}s=\frac{4}{5}$$ Let's use the same process to find cosine of t: $$\text{cos}^2 t + \text{sin}^2 t=1$$ Plug in for sin t: $$\text{cos}^2 t + \left(-\frac{12}{13}\right)^2=1$$ $$\text{cos}^2 t + \frac{144}{169}=1$$ $$\text{cos}^2 t=1 - \frac{144}{169}$$ $$\text{cos}^2 t=\frac{169}{169}- \frac{144}{169}$$ $$\text{cos}^2 t=\frac{25}{169}$$ Since t is in quadrant III, cosine is negative. Let's use the negative square root: $$\text{cos}\hspace{.1em}t=-\frac{5}{13}$$ Let's revisit our formula: $$\text{cos}(s + t)=\text{cos}\hspace{.1em}s \hspace{.1em}\text{cos}\hspace{.1em}t - \text{sin}\hspace{.1em}s \hspace{.1em}\text{sin}\hspace{.1em}t$$ Plug in and simplify: $$\text{cos}(s + t)=\text{cos}\hspace{.1em}s \hspace{.1em}\text{cos}\hspace{.1em}t - \text{sin}\hspace{.1em}s \hspace{.1em}\text{sin}\hspace{.1em}t$$ $$=\frac{4}{5}\cdot -\frac{5}{13}- \frac{3}{5}\cdot -\frac{12}{13}$$ $$=-\frac{20}{65}- \left(-\frac{36}{65}\right)$$ $$=-\frac{20}{65}+ \frac{36}{65}$$ $$=\frac{-20 + 36}{65}$$ $$=\frac{16}{65}$$

Skills Check:

Example #1

Find the exact value. $$\text{cos}\frac{11 π}{12}$$

Please choose the best answer.

A
$$\frac{\sqrt{6}- \sqrt{2}}{4}$$
B
$$\sqrt{3}- 2$$
C
$$\frac{-\sqrt{6}- \sqrt{2}}{4}$$
D
$$\frac{\sqrt{6}+ \sqrt{2}}{4}$$
E
$$\frac{\sqrt{3}+ 2}{4}$$

Example #2

Find the exact value. $$\text{cos}\frac{7π}{12}$$

Please choose the best answer.

A
$$\frac{\sqrt{6}+ \sqrt{2}}{4}$$
B
$$-2 - \sqrt{3}$$
C
$$\frac{-\sqrt{6}- \sqrt{2}}{4}$$
D
$$\sqrt{3}- 2$$
E
$$\frac{\sqrt{2}- \sqrt{6}}{4}$$

Example #3

Complete the identity. $$\text{sin}\hspace{.1em}θ=$$

Please choose the best answer.

A
$$\text{cos}\left(\frac{π}{12}+ θ\right)$$
B
$$\text{cos}\left(\frac{3π}{2}+ θ\right)$$
C
$$\text{cos}(θ + π)$$
D
$$\text{cos}(θ + \sqrt{π})$$
E
$$\text{cos}\hspace{.1em}2x$$
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