Chapter 4

Fill in the blanks. $$ \begin{array}{ll} \text { Function } & \text { Alternative Notation } & \text { Domain } & \text { Range } \end{array} $$ $$ y=\arcsin x \quad-\frac{\pi}{2} \leq y \leq \frac{\pi}{2} $$

Fill in the blanks. One period of a sine or cosine function is called one _______ of the sine or cosine curve.

Match the trigonometric function with its right triangle definition. (a) Sine (b) Cosine (c) Tangent (d) Cosecant (e) Secant (f) Cotangent (i) \(\frac{\text { hypotenuse }}{\text { adjacent }}\) (ii) \(\frac{\text { adjacent }}{\text { opposite }}\) (iii) \(\frac{\text { hypotenuse }}{\text { opposite }}\) (iv) \(\frac{\text { adjacent }}{\text { hypotenuse }}\) (v) \(\frac{\text { opposite }}{\text { hypotenuse }}\) (vi) \(\frac{\text { opposite }}{\text { adjacent }}\)

______ means "measurement of triangles."

Fill in the blanks. A _______ measures the acute angle a path or line of sight makes with a fixed north-south line.

Fill in the blanks. $$ \begin{array}{ll} \text { Function } & \text { Alternative Notation } & \text { Domain } & \text { Range } \end{array} $$ _________ $$ y=\cos ^{-1} x \quad-1 \leq x \leq 1 $$__________

Fill in the blanks. The graphs of the tangent, cotangent, secant, and cosecant functions all have ________ asymptotes.

Fill in the blanks. The ______ of a sine or cosine curve represents half the distance between the maximum and minimum values of the function.

Fill in the blanks. Relative to the angle \(\theta,\) the three sides of a right triangle are the _______ side, the ________ side, and the _______.

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

An ______ is determined by rotating a ray about its endpoint.

Fill in the blanks. A point that moves on a coordinate line is said to be in simple _______ _______ if 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. To sketch the graph of a secant or cosecant function, first make a sketch of its corresponding _______ function.

Fill in the blanks. For the function given by \(y=a \sin (b x-c), \frac{c}{b}\) represents the _______ ________ of the graph of the function.

Fill in the blanks. Cofunctions of _______ angles are equal.

The smallest number \(c\) for which a function \(f\) is periodic is called the ________ of \(f\)

Two angles that have the same initial and terminal sides are ______.

Fill in the blanks. The time for one complete cycle of a point in simple harmonic motion is its ______.

Fill in the blanks. For the functions given by \(f(x)=g(x) \cdot \sin x, g(x)\) is called the _______ factor of the function \(f(x)\).

Fill in the blanks. For the function given by \(y=d+a \cos (b x-c), d\) represents a ________ _______ of the graph of the function.

A function \(f\) is ______ if \(f(-t)=-f(t)\) and ______ if \(f(-t)=f(t) .\)

One _____ is the measure of a central angle that intercepts an arc equal to the radius of the circle.

Fill in the blanks. The number of cycles per second of a point in simple harmonic motion is its _______.

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

Fill in the blanks. The period of \(y=\tan x\) is _______.

Find the period and amplitude. $$ y=2 \sin 5 x $$

Angles that measure between 0 and \(\pi / 2\) are ______ angles, and angles that measure between \(\pi / 2\) and \(\pi\) are ______ angles.

Evaluate the expression without using a calculator. $$ \arcsin 0 $$

Fill in the blanks. The domain of \(y=\cot x\) is all real numbers such that ______.

Find the period and amplitude. $$ y=3 \cos 2 x $$

Two positive angles that have a sum of \(\pi / 2\) are ______ angles, whereas two positive angles that have a sum of \(\pi\) are ______ angles.

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

Fill in the blanks. The range of \(y=\sec x\) is ______.

Find the period and amplitude. $$ y=\frac{3}{4} \cos \frac{x}{2} $$

The angle measure that is equivalent to a rotation of \(\frac{1}{360}\) of a complete revolution about an angle's vertex is one ______.

Evaluate the expression without using a calculator. $$ \arccos 0 $$

Fill in the blanks. The period of \(y=\csc x\) is _______.

Find the period and amplitude. $$ y=-3 \sin \frac{x}{3} $$

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

180 degrees \(=\) ______ radians.

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

Find the period and amplitude. $$ y=\frac{1}{2} \sin \frac{\pi x}{3} $$

Find the point \((x, y)\) on the unit circle that corresponds to the real number \(t\). $$ t=\frac{\pi}{2} $$

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

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

Find the period and amplitude. $$ y=\frac{1}{2} \sin \frac{\pi x}{3} $$

Find the point \((x, y)\) on the unit circle that corresponds to the real number \(t\). $$ t=\pi $$

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

Find the period and amplitude. $$ y=-4 \sin x $$