Test Objectives
  • Demonstrate the ability to use a half-angle identity to find an exact value
  • Demonstrate the ability to find function values of s/2 given information about s
  • Demonstrate the ability to verify a trigonometric identity using half-angle identities
Half-Angle Identities Practice Test:

#1:

Instructions: Find the exact value.

$$a)\hspace{.1em}\text{sin}\hspace{.1em}345°$$

$$b)\hspace{.1em}\text{cos}\hspace{.1em}112.5°$$


#2:

Instructions: Find the exact value.

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

$$b)\hspace{.1em}\text{cos}\hspace{.1em}\frac{9π}{8}$$


#3:

Instructions: Find the exact value.

$$a)\hspace{.1em}\text{tan}\hspace{.1em}θ=\frac{4}{3}$$ $$180° < θ < 270°$$ Find: $$\text{cos}\hspace{.1em}\frac{θ}{2}$$

$$b)\hspace{.1em}\text{tan}\hspace{.1em}θ=\frac{1}{4}$$ $$180° < θ < 270°$$ Find: $$\text{tan}\hspace{.1em}\frac{θ}{2}$$


#4:

Instructions: Find the exact value.

$$a)\hspace{.1em}\text{sin}\hspace{.1em}θ=-\frac{4}{5}$$ $$180° < θ < 270°$$ Find: $$\text{sin}\hspace{.1em}\frac{θ}{2}$$

$$b)\hspace{.1em}\text{cos}\hspace{.1em}θ=-\frac{11}{17}$$ $$90° < θ < 180°$$ Find: $$\text{sin}\hspace{.1em}\frac{θ}{2}$$


#5:

Instructions: Verify each identity.

$$a)\hspace{.1em}\text{sec}^2 \frac{θ}{2}=\frac{2}{1 + \text{cos}\hspace{.1em}θ}$$

$$b) \hspace{.1em}\text{sin}^2 \frac{θ}{2}=\frac{\text{tan}\hspace{.1em}θ - \text{sin}\hspace{.1em}θ}{2 \hspace{.1em}\text{tan}\hspace{.1em}θ}$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}\text{sin}\hspace{.1em}345°=-\frac{\sqrt{2 - \sqrt{3}}}{2}$$

$$b)\hspace{.1em}\text{cos}\hspace{.1em}112.5°=-\frac{\sqrt{2 - \sqrt{2}}}{2}$$


#2:

Solutions:

$$a)\hspace{.1em}\text{sin}\hspace{.1em}\frac{π}{12}=\frac{\sqrt{2 - \sqrt{3}}}{2}$$

$$b)\hspace{.1em}\text{cos}\hspace{.1em}\frac{9π}{8}=-\frac{\sqrt{2 + \sqrt{2}}}{2}$$


#3:

Solutions:

$$a)\hspace{.1em}\text{cos}\hspace{.1em}\frac{θ}{2}=-\frac{\sqrt{5}}{5}$$

$$b)\hspace{.1em}\text{tan}\hspace{.1em}\frac{θ}{2}=-\sqrt{33 + 8\sqrt{17}}$$


#4:

Solutions:

$$a)\hspace{.1em}\text{sin}\hspace{.1em}\frac{θ}{2}=\frac{2\sqrt{5}}{5}$$

$$b)\hspace{.1em}\text{sin}\hspace{.1em}\frac{θ}{2}=\frac{\sqrt{238}}{17}$$


#5:

Solutions:

$$a)\hspace{.1em}\text{sec}^2 \frac{θ}{2}=\frac{2}{1 + \text{cos}\hspace{.1em}θ}$$ $$\frac{2}{1 + \text{cos}\hspace{.1em}θ}=\text{sec}^2 \frac{θ}{2}$$ $$=\frac{1}{\text{cos}^2 \large{\frac{θ}{2}}}$$ $$=\frac{1}{\left(\text{cos}\hspace{.1em}\large{\frac{θ}{2}}\right)^2}$$ $$=\frac{1}{\left(\pm \sqrt{\large{\frac{1 + \text{cos}\hspace{.1em}θ}{2}}}\right)^2}$$ $$=\large{\frac{1}{\large{\frac{1 + \text{cos}\hspace{.1em}θ}{2}}}}$$ $$=\frac{2}{1 + \text{cos}\hspace{.1em}θ}✓$$

$$b)\hspace{.1em}\hspace{.1em}\text{sin}^2 \frac{θ}{2}=\frac{\text{tan}\hspace{.1em}θ - \text{sin}\hspace{.1em}θ}{2 \hspace{.1em}\text{tan}\hspace{.1em}θ}$$ $$=\frac{\large{\frac{\text{sin}\hspace{.1em}θ}{\text{cos}\hspace{.1em}θ}- \text{sin}\hspace{.1em}θ}}{2\large{\frac{\text{sin}\hspace{.1em}θ}{\text{cos}\hspace{.1em}θ}}}$$ $$=\frac{\large{\frac{\text{sin}\hspace{.1em}θ}{\text{cos}\hspace{.1em}θ}- \text{sin}\hspace{.1em}θ}}{2\large{\frac{\text{sin}\hspace{.1em}θ}{\text{cos}\hspace{.1em}θ}}}\cdot \frac{\text{cos}\hspace{.1em}θ}{\text{cos}\hspace{.1em}θ}$$ $$=\frac{\text{sin}\hspace{.1em}θ - \text{sin}\hspace{.1em}θ \hspace{.1em}\text{cos}\hspace{.1em}θ}{2 \hspace{.1em}\text{sin}\hspace{.1em}θ}$$ $$=\frac{1 - \text{cos}\hspace{.1em}θ}{2}$$ $$=\left(\pm \sqrt{\frac{1 - \text{cos}\hspace{.1em}θ}{2}}\right)^2$$ $$=\left(\text{sin}\frac{θ}{2}\right)^2$$ $$=\text{sin}^2 \frac{θ}{2}✓$$