8.3 Inverse trigonometric functions  (Page 7/15)

Discuss why this statement is incorrect: $\arccos \left(\cos x\right)=x$ for all $x.$

Determine whether the following statement is true or false and explain your answer: $\arccos \left(-x\right)=\pi -\arccos x.$

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

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

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

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

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

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

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

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

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

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

$\arcsin \left(0.23\right)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\cos t$

$\sec t$

$\cot t$

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

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

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

Graph $y=\arccos x$ and state the domain and range of the function.

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

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

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

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 $\arctan \left(10,000\right).$ 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.