8.3 Inverse trigonometric functions  (Page 7/15)

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Discuss why this statement is incorrect: $\text{\hspace{0.17em}}\mathrm{arccos}\left(\mathrm{cos}\text{\hspace{0.17em}}x\right)=x\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}x.$

Determine whether the following statement is true or false and explain your answer: $\mathrm{arccos}\left(-x\right)=\pi -\mathrm{arccos}\text{\hspace{0.17em}}x.$

True . The angle, $\text{\hspace{0.17em}}{\theta }_{1}\text{\hspace{0.17em}}$ that equals $\text{\hspace{0.17em}}\mathrm{arccos}\left(-x\right)\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}x>0\text{\hspace{0.17em}}$ , will be a second quadrant angle with reference angle, $\text{\hspace{0.17em}}{\theta }_{2}\text{\hspace{0.17em}}$ , where $\text{\hspace{0.17em}}{\theta }_{2}\text{\hspace{0.17em}}$ equals $\text{\hspace{0.17em}}\mathrm{arccos}x$ , $x>0\text{\hspace{0.17em}}$ . Since $\text{\hspace{0.17em}}{\theta }_{2}\text{\hspace{0.17em}}$ is the reference angle for $\text{\hspace{0.17em}}{\theta }_{1}$ , ${\theta }_{2}=\pi -{\theta }_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{arccos}\left(-x\right)\text{\hspace{0.17em}}$ = $\text{\hspace{0.17em}}\pi -\mathrm{arccos}x$ -

Algebraic

For the following exercises, evaluate the expressions.

${\mathrm{sin}}^{-1}\left(\frac{\sqrt{2}}{2}\right)$

${\mathrm{sin}}^{-1}\left(-\frac{1}{2}\right)$

$-\frac{\pi }{6}$

${\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)$

${\mathrm{cos}}^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

$\frac{3\pi }{4}$

${\mathrm{tan}}^{-1}\left(1\right)$

${\mathrm{tan}}^{-1}\left(-\sqrt{3}\right)$

$-\frac{\pi }{3}$

${\mathrm{tan}}^{-1}\left(-1\right)$

${\mathrm{tan}}^{-1}\left(\sqrt{3}\right)$

$\frac{\pi }{3}$

${\mathrm{tan}}^{-1}\left(\frac{-1}{\sqrt{3}}\right)$

For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.

${\mathrm{cos}}^{-1}\left(-0.4\right)$

1.98

$\mathrm{arcsin}\left(0.23\right)$

$\mathrm{arccos}\left(\frac{3}{5}\right)$

0.93

${\mathrm{cos}}^{-1}\left(0.8\right)$

${\mathrm{tan}}^{-1}\left(6\right)$

1.41

For the following exercises, find the angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in the given right triangle. Round answers to the nearest hundredth.

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

${\mathrm{sin}}^{-1}\left(\mathrm{cos}\left(\pi \right)\right)$

${\mathrm{tan}}^{-1}\left(\mathrm{sin}\left(\pi \right)\right)$

0

${\mathrm{cos}}^{-1}\left(\mathrm{sin}\left(\frac{\pi }{3}\right)\right)$

${\mathrm{tan}}^{-1}\left(\mathrm{sin}\left(\frac{\pi }{3}\right)\right)$

0.71

${\mathrm{sin}}^{-1}\left(\mathrm{cos}\left(\frac{-\pi }{2}\right)\right)$

${\mathrm{tan}}^{-1}\left(\mathrm{sin}\left(\frac{4\pi }{3}\right)\right)$

-0.71

${\mathrm{sin}}^{-1}\left(\mathrm{sin}\left(\frac{5\pi }{6}\right)\right)$

${\mathrm{tan}}^{-1}\left(\mathrm{sin}\left(\frac{-5\pi }{2}\right)\right)$

$-\frac{\pi }{4}$

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(\frac{4}{5}\right)\right)$

$\mathrm{sin}\left({\mathrm{cos}}^{-1}\left(\frac{3}{5}\right)\right)$

0.8

$\mathrm{sin}\left({\mathrm{tan}}^{-1}\left(\frac{4}{3}\right)\right)$

$\mathrm{cos}\left({\mathrm{tan}}^{-1}\left(\frac{12}{5}\right)\right)$

$\frac{5}{13}$

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)\right)$

For the following exercises, find the exact value of the expression in terms of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ with the help of a reference triangle.

$\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(x-1\right)\right)$

$\frac{x-1}{\sqrt{-{x}^{2}+2x}}$

$\mathrm{sin}\left({\mathrm{cos}}^{-1}\left(1-x\right)\right)$

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(\frac{1}{x}\right)\right)$

$\frac{\sqrt{{x}^{2}-1}}{x}$

$\mathrm{cos}\left({\mathrm{tan}}^{-1}\left(3x-1\right)\right)$

$\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(x+\frac{1}{2}\right)\right)$

$\frac{x+0.5}{\sqrt{-{x}^{2}-x+\frac{3}{4}}}$

Extensions

For the following exercises, evaluate the expression without using a calculator. Give the exact value.

$\frac{{\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)-{\mathrm{cos}}^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\mathrm{sin}}^{-1}\left(\frac{\sqrt{3}}{2}\right)-{\mathrm{cos}}^{-1}\left(1\right)}{{\mathrm{cos}}^{-1}\left(\frac{\sqrt{3}}{2}\right)-{\mathrm{sin}}^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)-{\mathrm{sin}}^{-1}\left(0\right)}$

For the following exercises, find the function if $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t=\frac{x}{x+1}.$

$\mathrm{cos}\text{\hspace{0.17em}}t$

$\frac{\sqrt{2x+1}}{x+1}$

$\mathrm{sec}\text{\hspace{0.17em}}t$

$\mathrm{cot}\text{\hspace{0.17em}}t$

$\frac{\sqrt{2x+1}}{x}$

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(\frac{x}{x+1}\right)\right)$

${\mathrm{tan}}^{-1}\left(\frac{x}{\sqrt{2x+1}}\right)$

$t$

Graphical

Graph $\text{\hspace{0.17em}}y={\mathrm{sin}}^{-1}x\text{\hspace{0.17em}}$ and state the domain and range of the function.

Graph $\text{\hspace{0.17em}}y=\mathrm{arccos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and state the domain and range of the function.

domain $\text{\hspace{0.17em}}\left[-1,1\right];\text{\hspace{0.17em}}$ range $\text{\hspace{0.17em}}\left[0,\pi \right]\text{\hspace{0.17em}}$

Graph one cycle of $\text{\hspace{0.17em}}y={\mathrm{tan}}^{-1}x\text{\hspace{0.17em}}$ and state the domain and range of the function.

For what value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ does $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x={\mathrm{sin}}^{-1}x?\text{\hspace{0.17em}}$ Use a graphing calculator to approximate the answer.

approximately $\text{\hspace{0.17em}}x=0.00\text{\hspace{0.17em}}$

For what value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ does $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x={\mathrm{cos}}^{-1}x?\text{\hspace{0.17em}}$ Use a graphing calculator to approximate the answer.

Real-world applications

Suppose a 13-foot ladder is leaning against a building, reaching to the bottom of a second-ﬂoor window 12 feet above the ground. What angle, in radians, does the ladder make with the building?

Suppose you drive 0.6 miles on a road so that the vertical distance changes from 0 to 150 feet. What is the angle of elevation of the road?

An isosceles triangle has two congruent sides of length 9 inches. The remaining side has a length of 8 inches. Find the angle that a side of 9 inches makes with the 8-inch side.

Without using a calculator, approximate the value of $\text{\hspace{0.17em}}\mathrm{arctan}\left(10,000\right).\text{\hspace{0.17em}}$ Explain why your answer is reasonable.

A truss for the roof of a house is constructed from two identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side.

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