삼각함수를 포함한 방정식과 부등식

For the following exercises, find the exact value of the expression in terms of $x$ with the help of a reference triangle.

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

For the following exercises, find the exact value of the expression in terms of $x$ with the help of a reference triangle.

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

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

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

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

$\cos \,t$

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

$\sec \,t$

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

$\cot \,t$

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

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

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

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

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 $\arctan (10,000).$ Explain why your answer is reasonable.

A truss (interior beam structure) 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.

The line $y=\frac{3}{5}x$ passes through the origin in the x,y-plane. What is the measure of the angle that the line makes with the positive x-axis?

The line $y=\frac{-3}{7}x$ passes through the origin in the x,y-plane. What is the measure of the angle that the line makes with the negative x-axis?