Chapter 4

A _____ measures the acute angle that a path or line of sight makes with a fixed north-south line.

One period of a sine or cosine function is called one _____ of the sine or cosine curve.

Match each trigonometric function with its right triangle definition. $$\begin{array}{l}{\text { (a) sine }} \\ {\text { (i) } \frac{\text { hypotenuse }}{\text { adjacent }}}\end{array} $$ $$\begin{array}{l}{\text { (b) cosine }} \\ {\text { (ii) } \frac{\text { adjacent }}{\text { opposite }}}\end{array}$$$$ \begin{array}{c}{\text { (c) tangent }} \\ {\text { (iii) } \frac{\text { hypotenuse }}{\text { opposite }}}\end{array}$$$$\begin{array}{l}{\text { (d) cosecant }} \\ {\text { (iv) } \frac{\text { adjacent }}{\text { hypotenuse }}}\end{array}$$$$ \begin{array}{l}{\text { (e) secant }} \\ {\text { (v) } \frac{\text { opposite }}{\text { hypotenuse }}}\end{array} $$$$ \begin{array}{l}{\text { (f) cotangent }} \\ {\text { (vi) } \frac{\text { opposite }}{\text { adjacent }}}\end{array} $$

Fill in the blanks. Two angles that have the same initial and terminal sides are _______ .

Fill in the blanks. Function \(\quad\) Alternative Notation \(\quad\) Domain \(\quad\) Range _____ \(\quad\) \(y=\cos ^{-1} x\) \(\quad\) \(-1 \leq x \leq 1$$\quad\) _____

A point that moves on a coordinate line is said to be in simple _____ _____ when its distance \(d\) from the origin at time \(t\) is given by either \(d=a \sin \omega t\) or \(d=a \cos \omega t\).

Fill in the blanks. Relative to the acute angle \(\theta,\) the three sides of a right triangle are the ________side,the_____ side, and the______.

The graphs of the tangent, cotangent, secant, and cosecant functions have __________ asymptotes.

The _____ of a sine or cosine curve represents half the distance between the maximum and minimum values of the function.

A function \(f\) is __________________when there exists a positive real number \(c\) such that \(f(t+c)=f(t)\) for all \(t\) in the domain of \(f\).

Fill in the blanks. One _______ is the measure of a central angle that intercepts an arc equal to the radius of the circle.

Fill in the blanks. Function \(\quad\) Alternative Notation \(\quad\) Domain \(\quad\) Range \(y=\arctan x\) \(\quad\) _____ \(\quad\) _____ \(\quad\) _____

The time for one complete cycle of a point in simple harmonic motion is its _____.

To sketch the graph of a secant or cosecant function, first make a sketch of its ____________ function.

The smallest number \(c\) for which a function \(f\) is periodic is called the________________of \(f\).

Fill in the blanks. Two positive angles that have a sum of \(\pi / 2\) are _______ angles, whereas two positive angles that have a sum of \(\pi\) are _______ angles.

Fill in the blanks. Without restrictions, no trigonometric function has an _____ function.

The number of cycles per second of a point in simple harmonic motion is its _____.

An angle that measures from the horizontal upward to an object is called the angle of _____, whereas an angle that measures from the horizontal downward to an object is called the angle of __________.

For the function \(y=d+a \cos (b x-c), d\) represents a _____ _____ of the graph of the function.

For the function \(f(x)=g(x) \cdot \sin x, g(x)\) is called the __________ factor of the function \(f(x)\)

A function \(f\) is____________when \(f(-t)=-f(t)\) and_______________when \(f(-t)=f(t)\).

Fill in the blanks. The angle measure that is equivalent to a rotation of a complete revolution about an angle's vertex is one _______ .

Evaluate the expression without using a calculator. \(\arcsin \frac{1}{2}\)

Solving a Right Triangle, solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. $$A=30^{\circ}, \quad b=3$$

Vocabulary: Fill in the blanks. Let \(\theta\) be an angle in standard position with \((x, y)\) a point on the terminal side of \(\theta\) and \(r=\sqrt{x^{2}+y^{2}} \neq 0 .\) $$\frac{x}{r}=$$

Finding the Period and Amplitude, find the period and amplitude. $$ y=2 \sin 5 x $$

The period of \(y=\tan x\) is ____________

Fill in the blanks. The ______ speed of a particle is the ratio of the arc length to the time traveled, and the the ______ speed of a particle is the ratio of the central angle to the time traveled.

Evaluate the expression without using a calculator. \(\arcsin 0\)

Solving a Right Triangle, solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. $$B=54^{\circ}, \quad c=15$$

Finding the Period and Amplitude, find the period and amplitude. $$ y=3 \cos 2 x $$

Fill in the blanks. The area \(A\) of a sector of a circle with radius \(r\) and central angle \(\theta,\) where \(\theta\) is measured in radians, is given by the formula ______ .

Evaluate the expression without using a calculator. \(\arccos \frac{1}{2}\)

Solving a Right Triangle, solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. $$ B=71^{\circ}, \quad b=24 $$

Finding the Period and Amplitude, find the period and amplitude. $$ y=\frac{3}{4} \cos \frac{x}{2} $$

The range of \(y=\sec x\) is ____________.

Evaluate the expression without using a calculator. \(\arccos 0\)

The acute positive angle formed by the terminal side of an angle \(\theta\) and the horizontal axis is called the _____ angle of \(\theta\) and is denoted by \(\theta^{\prime}\).

Finding the Period and Amplitude, find the period and amplitude. $$ y=-3 \sin \frac{x}{3} $$

The period of \(y=\csc x\) is ______________.

Evaluate the expression without using a calculator. \(\arctan \frac{\sqrt{3}}{3}\)

Finding the Period and Amplitude, find the period and amplitude. $$ y=\frac{1}{2} \sin \frac{\pi x}{3} $$

Finding a Point on the Unit Circle In Exercises \(9-12,\) find the point \((x, y)\) on the unit circle that corresponds to the real number \(t\) . \(t=\pi / 2\)

Evaluate the expression without using a calculator. \(\arctan 1\)

Finding the Period and Amplitude, find the period and amplitude. $$ y=\frac{3}{2} \cos \frac{\pi x}{2} $$

Finding a Point on the Unit Circle In Exercises \(9-12,\) find the point \((x, y)\) on the unit circle that corresponds to the real number \(t\) . $$t=\pi / 4$$

Evaluate the expression without using a calculator. \(\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)

Finding the Period and Amplitude, find the period and amplitude. $$ y=-4 \sin x $$

Finding a Point on the Unit Circle In Exercises \(9-12,\) find the point \((x, y)\) on the unit circle that corresponds to the real number \(t\) . $$t=5 \pi / 6$$