Test Objectives
  • Demonstrate the ability to use the sum identity for cosine
  • Demonstrate the ability to use the difference identity for cosine
Sum & Difference Identities for Cosine Practice Test:

#1:

Instructions: Find the exact value.

$$a)\hspace{.1em}\text{cos}\hspace{.1em}\frac{17 π}{12}$$

$$b)\hspace{.1em}\text{cos}(-105°)$$


#2:

Instructions: Find the exact value.

$$a)\hspace{.1em}\text{cos}\frac{13π}{12}$$

$$b)\hspace{.1em}\text{cos}\hspace{.1em}285°$$


#3:

Instructions: Find the exact value.

$$a)\hspace{.1em}\text{cos}\frac{2π}{9}\hspace{.1em}\text{cos}\frac{4π}{9}- \text{sin}\frac{2π}{9}\hspace{.1em}\text{sin}\frac{4π}{9}$$

$$b)\hspace{.1em}\text{cos}\frac{25π}{18}\hspace{.1em}\text{cos}\frac{5π}{9}+ \text{sin}\frac{25π}{18}\hspace{.1em}\text{sin}\frac{5π}{9}$$


#4:

Instructions: verify each identity.

$$a)\hspace{.1em}\text{cos}\left(θ - \frac{3π}{2}\right)=-\text{sin}\hspace{.1em}θ$$

$$b)\hspace{.1em}\text{sec}\left(\frac{3 π}{2}- x\right)=-\text{csc}\hspace{.1em}x$$


#5:

Instructions: Find cos(s - t).

$$a)\hspace{.1em}\text{cos}\hspace{.1em}s=-\frac{1}{5}, \text{sin}\hspace{.1em}t=\frac{3}{5}$$ $$\text{s, t in QII}$$

$$b)\hspace{.1em}\text{sin}\hspace{.1em}s=\frac{\sqrt{5}}{7}, \text{sin}\hspace{.1em}t=\frac{\sqrt{6}}{8}$$ $$\text{s, t in QI}$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}\frac{\sqrt{2}- \sqrt{6}}{4}$$

$$b)\hspace{.1em}\frac{\sqrt{2}- \sqrt{6}}{4}$$


#2:

Solutions:

$$a)\hspace{.1em}\frac{-\sqrt{6}- \sqrt{2}}{4}$$

$$b)\hspace{.1em}\frac{\sqrt{6}- \sqrt{2}}{4}$$


#3:

Solutions:

$$a)\hspace{.1em}-\frac{1}{2}$$

$$b)\hspace{.1em}-\frac{\sqrt{3}}{2}$$


#4:

Solutions:

$$a)\hspace{.1em}\hspace{.1em}\text{cos}\left(θ - \frac{3π}{2}\right)=-\text{sin}\hspace{.1em}θ$$ $$-\text{sin}\hspace{.1em}θ=\text{cos}\left(θ - \frac{3π}{2}\right)$$ $$=\text{cos}\hspace{.1em}θ \hspace{.1em}\text{cos}\hspace{.1em}\frac{3π}{2}+ \text{sin}\hspace{.1em}θ \hspace{.1em}\text{sin}\hspace{.1em}\frac{3π}{2}$$ $$=\text{cos}\hspace{.1em}θ \cdot 0 + \text{sin}\hspace{.1em}θ \cdot -1$$ $$=-\text{sin}\hspace{.1em}θ ✓$$

$$b)\hspace{.1em}\text{sec}\left(\frac{3 π}{2}- x\right)=-\text{csc}\hspace{.1em}x$$ $$-\text{csc}\hspace{.1em}x=\text{sec}\left(\frac{3 π}{2}- x\right)$$ $$=\frac{1}{\text{cos}\left(\frac{3 π}{2}- x\right)}$$ $$=\frac{1}{\text{cos}\frac{3π}{2}\text{cos}\hspace{.1em}x + \text{sin}\frac{3π}{2}\text{sin}\hspace{.1em}x}$$ $$=\frac{1}{0 \cdot \text{cos}\hspace{.1em}x + (-1) \cdot \text{sin}\hspace{.1em}x}$$ $$=-\frac{1}{\text{sin}\hspace{.1em}x}$$ $$=-\text{csc}\hspace{.1em}x$$


#5:

Solutions:

$$a)\hspace{.1em}\frac{4 + 6\sqrt{6}}{25}$$

$$b)\hspace{.1em}\frac{2\sqrt{638}+ \sqrt{30}}{56}$$