- Demonstrate the ability to find the exact value of a trigonometric expression
- Demonstrate the ability to simplify a trigonometric expression
- Demonstrate the ability to verify a trigonometric identity
#1:
Instructions: Derive the given identities.
$$a)\hspace{.1em}\text{sin}(A + B)$$
$$b)\hspace{.1em}\text{sin}(A - B)$$
$$c)\hspace{.1em}\text{tan}(A + B)$$
$$d)\hspace{.1em}\text{tan}(A - B)$$
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#2:
Instructions: Find the exact value.
$$a)\hspace{.1em}\text{sin}\frac{5 π}{12}$$
$$b)\hspace{.1em}\text{tan}\hspace{.1em}255 °$$
$$c)\hspace{.1em}\text{sin}\frac{47 π}{18}\hspace{.1em}\text{cos}\frac{10π}{9}- \text{cos}\frac{47 π}{18}\hspace{.1em}\text{sin}\frac{10π}{9}$$
$$d)\hspace{.1em}\frac{\text{tan}\large{\frac{π}{18}}+ \text{tan}\large{\frac{5 π}{18}}}{1 - \text{tan}\large{\frac{π}{18}}\text{tan}\large{\frac{5 π}{18}}}$$
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#3:
Instructions: Simplify.
$$a)\hspace{.1em}\text{sin}(-4v)\hspace{.1em}\text{cos}\hspace{.1em}5v + \text{cos}(-4v)\hspace{.1em}\text{sin}\hspace{.1em}5v$$
$$b)\hspace{.1em}\frac{\text{tan}\hspace{.1em}θ - \text{tan}\hspace{.1em}4θ}{1 + \text{tan}\hspace{.1em}θ \hspace{.1em}\text{tan}\hspace{.1em}4θ}$$
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#4:
Instructions: Find sin(θ + β), tan(θ + β), and the quadrant of θ + β.
$$a)\hspace{.1em}\text{cos}\hspace{.1em}θ=\frac{3}{5}$$ $$\text{sin}\hspace{.1em}β=\frac{5}{13}$$ θ and β are in quadrant I
$$b)\hspace{.1em}\text{cos}\hspace{.1em}θ=-\frac{8}{17}$$ $$\text{cos}\hspace{.1em}β=-\frac{3}{5}$$ θ and β are in quadrant III
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#5:
Instructions: Verify each identity.
$$a)\hspace{.1em}\frac{\text{tan}\hspace{.1em}θ + 1}{1 - \text{tan}\hspace{.1em}θ}=\text{tan}(θ - 135°)$$
$$b) \hspace{.1em}\frac{\text{cos}(θ - β)}{\text{cos}\hspace{.1em}θ\hspace{.1em}\text{sin}\hspace{.1em}β}=\text{tan}\hspace{.1em}θ + \text{cot}\hspace{.1em}β$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.1em}$$ $$\text{sin}(A + B)=\text{cos}(90° - (A + B))$$ $$=\text{cos}((90° - A) - B)$$ $$=\text{cos}\hspace{.1em}(90° - A)\hspace{.1em}\text{cos}\hspace{.1em}B + \text{sin}\hspace{.1em}(90° - A) \hspace{.1em}\text{sin}\hspace{.1em}B$$ $$=\text{sin}\hspace{.1em}A\hspace{.1em}\text{cos}\hspace{.1em}B + \text{cos}\hspace{.1em}A\hspace{.1em}\text{sin}\hspace{.1em}B$$
$$b)\hspace{.1em}$$ $$\text{sin}(A - B)=\text{sin}(A + (-B))$$ $$=\text{sin}\hspace{.1em}A\hspace{.1em}\text{cos}(-B)+\text{cos}\hspace{.1em}A\hspace{.1em}\text{sin}(-B)$$ $$=\text{sin}\hspace{.1em}A\hspace{.1em}\text{cos}\hspace{.1em}B-\text{cos}\hspace{.1em}A\hspace{.1em}\text{sin}\hspace{.1em}B$$
$$c)\hspace{.1em}$$ $$\text{tan}(A + B)=\frac{\text{sin}(A + B)}{\text{cos}(A + B)}$$ $$=\frac{\text{sin}\hspace{.1em}A\hspace{.1em}\text{cos}\hspace{.1em}B + \text{cos}\hspace{.1em}A\hspace{.1em}\text{sin}\hspace{.1em}B}{\text{cos}\hspace{.1em}A\hspace{.1em}\text{cos}\hspace{.1em}B - \text{sin}\hspace{.1em}A\hspace{.1em}\text{sin}\hspace{.1em}B}$$ $$=\frac{\text{sin}\hspace{.1em}A\hspace{.1em}\text{cos}\hspace{.1em}B + \text{cos}\hspace{.1em}A\hspace{.1em}\text{sin}\hspace{.1em}B}{\text{cos}\hspace{.1em}A\hspace{.1em}\text{cos}\hspace{.1em}B - \text{sin}\hspace{.1em}A\hspace{.1em}\text{sin}\hspace{.1em}B}\cdot \large{\frac{\frac{1}{\text{cos}\hspace{.1em}A \hspace{.1em}\text{cos}\hspace{.1em}B}}{\frac{1}{\text{cos}\hspace{.1em}A \hspace{.1em}\text{cos}\hspace{.1em}B}}}$$ $$=\Large{\frac{\frac{\text{sin}\hspace{.1em}A\hspace{.1em}\text{cos}\hspace{.1em}B + \text{cos}\hspace{.1em}A\hspace{.1em}\text{sin}\hspace{.1em}B}{\text{cos}\hspace{.1em}A \hspace{.1em}\text{cos}\hspace{.1em}B}}{\frac{\text{cos}\hspace{.1em}A\hspace{.1em}\text{cos}\hspace{.1em}B - \text{sin}\hspace{.1em}A\hspace{.1em}\text{sin}\hspace{.1em}B}{\text{cos}\hspace{.1em}A \hspace{.1em}\text{cos}\hspace{.1em}B}}}$$ $$=\Large{\frac{\frac{\text{sin}\hspace{.1em}A\hspace{.1em}\text{cos}\hspace{.1em}B}{\text{cos}\hspace{.1em}A \hspace{.1em}\text{cos}\hspace{.1em}B}+ \frac{\text{cos}\hspace{.1em}A\hspace{.1em}\text{sin}\hspace{.1em}B}{\text{cos}\hspace{.1em}A \hspace{.1em}\text{cos}\hspace{.1em}B}}{\frac{\text{cos}\hspace{.1em}A\hspace{.1em}\text{cos}\hspace{.1em}B}{\text{cos}\hspace{.1em}A \hspace{.1em}\text{cos}\hspace{.1em}B}- \frac{\text{sin}\hspace{.1em}A\hspace{.1em}\text{sin}\hspace{.1em}B}{\text{cos}\hspace{.1em}A \hspace{.1em}\text{cos}\hspace{.1em}B}}}$$ $$=\Large{\frac{\frac{\text{sin}\hspace{.1em}A}{\text{cos}\hspace{.1em}A}+ \frac{\text{sin}\hspace{.1em}B}{\text{cos}\hspace{.1em}B}}{1 - \frac{\text{sin}\hspace{.1em}A}{\text{cos}\hspace{.1em}A}\cdot \frac{\text{sin}\hspace{.1em}B}{\text{cos}\hspace{.1em}B}}}$$ $$=\frac{\text{tan}\hspace{.1em}A + \text{tan}\hspace{.1em}B}{1 - \text{tan}\hspace{.1em}A \hspace{.1em}\text{tan}\hspace{.1em}B}$$
$$d)\hspace{.1em}$$ $$\text{tan}(A - B)=\text{tan}(A + (-B))$$ $$=\frac{\text{tan}\hspace{.1em}A + \text{tan}(-B)}{1 - \text{tan}\hspace{.1em}A\hspace{.1em}\text{tan}(-B)}$$ $$=\frac{\text{tan}\hspace{.1em}A - \text{tan}\hspace{.1em}B}{1 + \text{tan}\hspace{.1em}A\hspace{.1em}\text{tan}\hspace{.1em}B}$$
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#2:
Solutions:
$$a)\hspace{.1em}\frac{\sqrt{6}+ \sqrt{2}}{4}$$
$$b)\hspace{.1em}2 + \sqrt{3}$$
$$c)\hspace{.1em}{-}1$$
$$d)\hspace{.1em}\sqrt{3}$$
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#3:
Solutions:
$$a)\hspace{.1em}\text{sin}\hspace{.1em}v$$
$$b)\hspace{.1em}{-}\text{tan}\hspace{.1em}3θ$$
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#4:
Solutions:
$$a)\hspace{.1em}\text{sin}(θ + β)=\frac{63}{65}$$ $$\text{tan}(θ + β)=\frac{63}{16}$$ (θ + β) is in quadrant I
$$b)\hspace{.1em}\text{sin}(θ + β)=\frac{77}{85}$$ $$\text{tan}(θ + β)=-\frac{77}{36}$$ (θ + β) is in quadrant II
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#5:
Solutions:
$$a)\hspace{.1em}\frac{\text{tan}\hspace{.1em}θ + 1}{1 - \text{tan}\hspace{.1em}θ}=\text{tan}(θ - 135°)$$ $$=\text{tan}(θ - 135°)$$ $$=\frac{\text{tan}\hspace{.1em}θ - (-1)}{1 + \text{tan}\hspace{.1em}θ\hspace{.1em}(-1)}$$ $$=\frac{\text{tan}\hspace{.1em}θ + 1}{1 - \text{tan}\hspace{.1em}θ}✓$$
$$b)\hspace{.1em}\frac{\text{cos}(θ - β)}{\text{cos}\hspace{.1em}θ\hspace{.1em}\text{sin}\hspace{.1em}β}=\text{tan}\hspace{.1em}θ + \text{cot}\hspace{.1em}β$$ $$\text{tan}\hspace{.1em}θ + \text{cot}\hspace{.1em}β=\frac{\text{cos}(θ - β)}{\text{cos}\hspace{.1em}θ\hspace{.1em}\text{sin}\hspace{.1em}β}$$ $$=\frac{\text{cos}\hspace{.1em}θ \hspace{.1em}\text{cos}β + \text{sin}\hspace{.1em}θ \hspace{.1em}\text{sin}\hspace{.1em}β}{\text{cos}\hspace{.1em}θ\hspace{.1em}\text{sin}\hspace{.1em}β}$$ $$=\frac{\text{cos}\hspace{.1em}θ \hspace{.1em}\text{cos}β}{\text{cos}\hspace{.1em}θ\hspace{.1em}\text{sin}\hspace{.1em}β}+ \frac{\text{sin}\hspace{.1em}θ \hspace{.1em}\text{sin}\hspace{.1em}β}{\text{cos}\hspace{.1em}θ\hspace{.1em}\text{sin}\hspace{.1em}β}$$ $$=\frac{\text{cos}β}{\text{sin}\hspace{.1em}β}+ \frac{\text{sin}\hspace{.1em}θ}{\text{cos}\hspace{.1em}θ}$$ $$=\text{cot}\hspace{.1em}β+ \text{tan}\hspace{.1em}θ$$ $$=\text{tan}\hspace{.1em}θ+ \text{cot}\hspace{.1em}β✓$$
