- Demonstrate the ability to evaluate trigonometric function composition
- Demonstrate the ability to solve trigonometric equations using inverses
- Demonstrate the ability to solve trigonometric equations involving inverse trigonometric functions
#1:
Instructions: Find the exact value.
$$a)\hspace{.1em}\text{tan}^{-1}\left(\text{tan}\left(\frac{7π}{3}\right)\right)$$
$$b)\hspace{.1em}\text{sin}^{-1}\left(\text{sin}\left(-\frac{17π}{6}\right)\right)$$
$$c)\hspace{.1em}\text{cos}^{-1}\left(\text{cos}\left(-\frac{7π}{5}\right)\right)$$
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#2:
Instructions: Solve each equation for 0 ≤ x < 2π.
Try solving these equations using inverses.
$$a)\hspace{.1em}{-}2\text{csc}\left(\frac{x}{3}+ \frac{3π}{4}\right)=-4$$
$$b)\hspace{.1em}4\text{cos}\left(\frac{x}{2}+ \frac{π}{4}\right)=-2\sqrt{3}$$
$$c)\hspace{.1em}6\text{csc}\left(3x + \frac{5π}{6}\right)=-4\sqrt{3}$$
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#3:
Instructions: Solve each equation for exact solutions.
$$a)\hspace{.1em}4\hspace{.1em}\text{sin}^{-1}(x)=π$$
$$b)\hspace{.1em}\text{sin}^{-1}\left(x - \frac{π}{3}\right)=\frac{π}{6}$$
$$c)\hspace{.1em}\frac{4}{7}\text{cos}^{-1}\left(\frac{x}{4}\right)=π$$ Hint: Consider the range of arccos, check all solutions.
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#4:
Instructions: Solve each equation for exact solutions.
$$a)\hspace{.1em}\text{cos}^{-1}\frac{5}{13}=\text{tan}^{-1}(x)$$
$$b)\hspace{.1em}\text{sin}^{-1}\frac{8}{17}=\text{cos}^{-1}(x)$$
$$c)\hspace{.1em}\text{sec}^{-1}\frac{13}{12}=\text{cot}^{-1}(x)$$
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#5:
Instructions: Solve each equation for exact solutions.
These identities will speed up your work. Try solving both ways.
$$\text{cos}^{-1}(x) + \text{sin}^{-1}(x)=\frac{π}{2}$$
$$\text{tan}^{-1}(x)=\text{sin}^{-1}\left(\frac{x}{\sqrt{x^2 + 1}}\right)$$
$$a)\hspace{.1em}\text{cos}^{-1}(x) - \text{sin}^{-1}(x)=\frac{7π}{6}$$
$$b)\hspace{.1em}\text{cos}^{-1}(x) + \text{tan}^{-1}(x)=\frac{π}{2}$$
$$c)\hspace{.1em}2\text{sin}^{-1}(x) + \text{cos}^{-1}(x)=π$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.1em}\frac{π}{3}$$
$$b)\hspace{.1em}{-}\frac{π}{6}$$
$$c)\hspace{.1em}\frac{3π}{5}$$
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#2:
Solutions:
$$a)\hspace{.1em}x=\frac{π}{4}$$
$$b)\hspace{.1em}x=\frac{7π}{6}, \frac{11π}{6}$$
$$c)\hspace{.1em}x=\frac{π}{6}, \frac{5π}{18}, \frac{5π}{6}, \frac{17π}{18}, \frac{3π}{2}, \frac{29π}{18}$$
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#3:
Solutions:
$$a)\hspace{.1em}x=\frac{\sqrt{2}}{2}$$
$$b)\hspace{.1em}x=\frac{1}{2}+ \frac{π}{3}$$
$$c)\hspace{.1em}\text{No Solution}$$
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#4:
Solutions:
$$a)\hspace{.1em}x=\frac{12}{5}$$
$$b)\hspace{.1em}x=\frac{15}{17}$$
$$c)\hspace{.1em}x=\frac{12}{5}$$
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#5:
Solutions:
$$a)\hspace{.1em}x=-\frac{\sqrt{3}}{2}$$
$$b)\hspace{.1em}x=0$$
$$c)\hspace{.1em}x=1$$