Chapter 5

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

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

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

Match each trigonometric function with its right triangle definition. (a) sine (b) cosine (c) tangent (d) cosecant (e) secant (f) cotangent (i) \(\frac{\text { hyp }}{\text { adj }}\) (ii) \(\frac{\text { opp }}{\text { adj }}\) (iii) \(\frac{\text { opp }}{\text { hyp }}\) (iv) \(\frac{\text { adj }}{\text { opp }}\) (v) \(\frac{\text { hyp }}{\text { opp }}\) (vi) \(\frac{\text { adj }}{\text { hyp }}\)

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

Fill in the blank. _______ means “measurement of triangles.”

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

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

One period of a sine function is called __________ of the sine curve.

Fill in the blank. \(\mathrm{A}(\mathrm{n})\) _______ is determined by rotating a ray about its endpoint.

Fill in the blank. Does the bearing of \(\mathrm{N} 20^{\circ} \mathrm{E}\) mean 20 degrees north of east?

What notation can you use to represent the inverse sine function?

Which two parent trigonometric functions have a period of \(\pi\) and a range that consists of the set of all real numbers?

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

Fill in the blanks. 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 _____.

The period of a sine or cosine function is given by _________.

Fill in the blank. An angle with its initial side coinciding with the positive \(x\) -axis and the origin as its vertex is said to be in _______ .

Fill in the blank. What is the amplitude of the simple harmonic motion described by \(d=3 \sin \frac{\pi}{2} r ?\)

Does arccos \(x=\frac{1}{\cos x} ?\)

What is the damping factor of the function \(f(x)=e^{2 x} \sin x ?\)

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

For the equation \(y=a \sin (b x-c), \frac{c}{b}\) is the ____________ of the graph of the equation.

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

Use the graph of the function to answer each question. (a) Find any \(x\) -intercepts of the graph of \(y=f(x)\). (b) Find any \(y\) -intercepts of the graph of \(y=f(x)\). (c) Find the intervals on which the graph of \(y=f(x)\) is increasing and the intervals on which the graph of \(y=f(x)\) is decreasing. (d) Find all relative extrema, if any, of the graph of \(y=f(x)\) (e) Find all vertical asymptotes, if any, of the graph of \(y=f(x)\) \(f(x)=\tan x\)

Find the exact value of each expression, if possible, without using a calculator. (a) \(\arcsin \frac{1}{2}\) (b) arcsin 0

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

What is the period of the sine function \(y=\sin x ?\)

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

Use the graph of the function to answer each question. (a) Find any \(x\) -intercepts of the graph of \(y=f(x)\). (b) Find any \(y\) -intercepts of the graph of \(y=f(x)\). (c) Find the intervals on which the graph of \(y=f(x)\) is increasing and the intervals on which the graph of \(y=f(x)\) is decreasing. (d) Find all relative extrema, if any, of the graph of \(y=f(x)\) (e) Find all vertical asymptotes, if any, of the graph of \(y=f(x)\) \(f(x)=\cot x\)

Find the exact value of each expression, if possible, without using a calculator. (a) \(\arccos \frac{1}{2}\) (b) arccos 0

How do you find the period of a cosine function of the form \(y=\cos b x ?\)

Fill in the blank. The _______ speed of a particle is a ratio of the change in the central angle to the time.

Find the exact value of each expression, if possible, without using a calculator. (a) \(\arcsin (-1)\) (b) arccos 1

Use the graph of the function to answer each question. (a) Find any \(x\) -intercepts of the graph of \(y=f(x)\). (b) Find any \(y\) -intercepts of the graph of \(y=f(x)\). (c) Find the intervals on which the graph of \(y=f(x)\) is increasing and the intervals on which the graph of \(y=f(x)\) is decreasing. (d) Find all relative extrema, if any, of the graph of \(y=f(x)\) (e) Find all vertical asymptotes, if any, of the graph of \(y=f(x)\) \(f(x)=\sec x\)

Fill in the blank. A function \(f\) is _____ when \(f(-t)=-f(t)\).

Describe the effect of the constant \(d\) on the graph of \(y=\sin x+d.\)

Is one-half revolution of a circle equal to \(90^{\circ}\) or \(180^{\circ} ?\)

Use the graph of the function to answer each question. (a) Find any \(x\) -intercepts of the graph of \(y=f(x)\). (b) Find any \(y\) -intercepts of the graph of \(y=f(x)\). (c) Find the intervals on which the graph of \(y=f(x)\) is increasing and the intervals on which the graph of \(y=f(x)\) is decreasing. (d) Find all relative extrema, if any, of the graph of \(y=f(x)\) (e) Find all vertical asymptotes, if any, of the graph of \(y=f(x)\) \(f(x)=\csc x\)

Find the exact value of each expression, if possible, without using a calculator. (a) arctan 1 (b) \(\arccos (-1)\)

Fill in the blank. A function \(f\) is _____ when \(f(-t)=f(t)\).

What is the amplitude of \(y=-4.5 \sin x ?\)

What is the sum of two complementary angles in degrees? in radians?

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\tan \frac{x}{5}\)

What do you call the acute angle formed by the terminal side of an angle \(\theta\) in standard position and the horizontal axis?

Are the angles \(315^{\circ}\) and \(-225^{\circ}\) coterminal?

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\tan 3 x\)

In which quadrants is \(\cos \theta\) positive?

Use the graph of the function to answer each question. (a) Find the \(x\) -intercepts of the graph of \(y=f(x).\) (b) Find the \(y\) -intercept of the graph of \(y=f(x).\) (c) Find the intervals on which the graph of \(y=f(x)\) is increasing and the intervals on which the graph of \(y=f(x)\) is decreasing. (d) Find the relative extrema of the graph of \(y=f(x).\) $$f(x)=\cos x$$

Is the angle \(\frac{2 \pi}{3}\) acute or obtuse?

Find the exact value of each expression, if possible, without using a calculator. (a) \(\arctan (-\sqrt{3})\) (b) \(\arccos \frac{\sqrt{3}}{2}\)