More Basic Functions and Graphs: Get Stronger

Trigonometric Functions

For the following exercises (1-8), convert each angle in degrees to radians. Write the answer as a multiple of [latex]\pi[/latex].

  1. [latex]240°[/latex] 
  2. [latex]-60°[/latex] 
  3. [latex]330°[/latex] 

For the following exercises (4-5), convert each angle in radians to degrees.

  1. [latex]\large \frac{7\pi}{6}[/latex] rad
  2. [latex]-3\pi[/latex] rad

Evaluate the following functional values (6-8).

  1. [latex]\cos \Big(\large\frac{4\pi}{3}\Big)[/latex]
  2. [latex]\sin\left(-\large\frac{3\pi}{4}\right)[/latex]
  3. [latex]\sin\Big(\large\frac{\pi}{12}\Big)[/latex]

For the following exercises (9-11), consider triangle [latex]ABC[/latex], a right triangle with a right angle at [latex]C[/latex]. 

  1. Find the missing side of the triangle
  2. Find the six trigonometric function values for the angle at [latex]A[/latex].

Where necessary, round to one decimal place.

An image of a triangle. The three corners of the triangle are labeled “A”, “B”, and “C”. Between the corner A and corner C is the side b. Between corner C and corner B is the side a. Between corner B and corner A is the side c. The angle of corner C is marked with a right triangle symbol. The angle of corner A is marked with an angle symbol.

  1. [latex]a=4, \, c=7[/latex]
  2. [latex]a=85.3, \, b=125.5[/latex]
  3. [latex]a=84, \, b=13[/latex]

For the following exercises (12-13), [latex]P[/latex] is a point on the unit circle.

  1. find the (exact) missing coordinate value of each point
  2. find the values of the six trigonometric functions for the angle [latex]\theta[/latex] with a terminal side that passes through point [latex]P[/latex].

Rationalize all denominators.

  1. [latex]P\left(\frac{7}{25},y\right), \, y>0[/latex]
  2. [latex]P\left(x,\frac{\sqrt{7}}{3}\right), \, x<0[/latex]

For the following exercises (14-17), simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

  1. [latex]\tan^2 x+\sin x\csc x[/latex]
  2. [latex]\dfrac{\tan^2 x}{\sec^2 x}[/latex]
  3. [latex](1+\tan \theta)^2-2\tan \theta[/latex]
  4. [latex]\dfrac{\cos t}{\sin t} + \dfrac{\sin t}{1+\cos t}[/latex]

For the following exercises (18-21), verify that each equation is an identity.

  1. [latex]\dfrac{\tan \theta \cot \theta}{\csc \theta} =\sin \theta[/latex]
  2. [latex]\dfrac{\sin t}{\csc t} + \dfrac{\cos t}{\sec t} =1[/latex]
  3. [latex]\cot \gamma + \tan \gamma = \sec \gamma \csc \gamma[/latex]
  4. [latex]\dfrac{1}{1-\sin \alpha} + \dfrac{1}{1+\sin \alpha } =2\sec^2 \alpha[/latex]

For the following exercises (22-25), solve the trigonometric equations on the interval [latex]0\le \theta <2\pi[/latex].

  1. [latex]2\sin \theta -1=0[/latex]
  2. [latex]2\tan^2 \theta =2[/latex]
  3. [latex]\sqrt{3}\cot \theta +1=0[/latex]
  4. [latex]2\cos \theta \sin \theta =\sin \theta[/latex]

For the following exercises (26-27), each graph is of the form [latex]y=A\sin Bx[/latex]  or  [latex]y=A\cos Bx[/latex], where [latex]B>0[/latex]. Write the equation of the graph.

  1. An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function that starts at the point (-4, 0) and decreases until the point (-2, 4). After this point the function begins increasing until it hits the point (2, 4). After this point the function begins decreasing again. The x intercepts of the function on this graph are at (-4, 0), (0, 0), and (4, 0). The y intercept is at the origin.
  2. An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function. There are many periods and only a few will be explained. The function begins decreasing at the point (-1, 1) and decreases until the point (-0.5, -1). After this point the function increases until it hits the point (0, 1). After this point the function decreases until it hits the point (0.5, -1). After this point the function increases until it hits the point (1, 1). After this point the function decreases again. The x intercepts of the function on this graph are at (-0.75, 0), (-0.25, 0), (0.25, 0), and (0.75, 0). The y intercept is at (0, 1).

For the following exercises (28-30) find the,

  1. amplitude
  2. period
  3. phase shift with direction for each function.
  1. [latex]y=\sin\left(x-\dfrac{\pi}{4}\right)[/latex]
  2. [latex]y=\frac{-1}{2}\sin\left(\frac{1}{4}x\right)[/latex]
  3. [latex]y=-3\sin(\pi x+2)[/latex]

For the following exercises (31-35), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.

  1. The diameter of a wheel rolling on the ground is [latex]40[/latex] in. If the wheel rotates through an angle of [latex]120^{\circ}[/latex], how many inches does it move? Approximate to the nearest whole inch.
  2. As a point [latex]P[/latex] moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, [latex]\omega[/latex], and is given by [latex]\omega =\dfrac{\theta}{t}[/latex], where [latex]\theta[/latex] is in radians and [latex]t[/latex] is time. Find the angular speed for the given data. Round to the nearest thousandth.

    a. [latex]\theta =\frac{7\pi}{4}[/latex] rad, [latex]t=10[/latex] sec
    b. [latex]\theta =\frac{3\pi }{5}[/latex] rad, [latex]t=8[/latex] sec
    c. [latex]\theta =\frac{2\pi }{9}[/latex] rad, [latex]t=1[/latex] min
    d. [latex]\theta =23.76[/latex] rad, [latex]t=14[/latex] min

  3. The area of an isosceles triangle with equal sides of length [latex]x[/latex] is

    [latex]\frac{1}{2}x^2 \sin \theta[/latex],

    where [latex]\theta[/latex] is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle [latex]\theta =\frac{5\pi}{12}[/latex] rad.

  4. An alternating current for outlets in a home has voltage given by the function

    [latex]V(t)=150\cos 368t[/latex],

    where [latex]V[/latex] is the voltage in volts at time [latex]t[/latex] in seconds.

    1. Find the period of the function and interpret its meaning.
    2. Determine the number of periods that occur when 1 sec has passed.
  5. Suppose that [latex]T=50+10\sin\left[\frac{\pi}{12}(t-8)\right][/latex] is a mathematical model of the temperature (in degrees Fahrenheit) at [latex]t[/latex] hours after midnight on a certain day of the week.
    1. Determine the amplitude and period.
    2. Find the temperature [latex]7[/latex] hours after midnight.
    3. At what time does [latex]T=60^{\circ}[/latex]?
    4. Sketch the graph of [latex]T[/latex] over [latex]0\le t\le 24[/latex].

Inverse Functions

For the following exercises (1-3), use the horizontal line test to determine whether each of the given graphs is one-to-one.

  1. An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a function that decreases in a straight in until the origin, where it begins to increase in a straight line. The x intercept and y intercept are both at the origin.
  2. An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a function that resembles a semi-circle, the top half of a circle. The function starts at the point (-3, 0) and increases until the point (0, 3), where it begins decreasing until it ends at the point (3, 0). The x intercepts are at (-3, 0) and (3, 0). The y intercept is at (0, 3).
  3. An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a curved function that is always increasing. The x intercept and y intercept are both at the origin.

For the following exercises (4-6),

  1. find the inverse function
  2. find the domain and range of the inverse function.
  1. [latex]f(x)=x^2-4, \, x \ge 0[/latex]
  2. [latex]f(x)=x^3+1[/latex]
  3. [latex]f(x)=\sqrt{x-1}[/latex]

For the following exercises (7-8), use the graph of [latex]f[/latex] to sketch the graph of its inverse function.

  1. An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of an increasing straight line function labeled “f” that is always increasing. The x intercept is at (-2, 0) and y intercept are both at (0, 1).
  2. An image of a graph. The x axis runs from -8 to 8 and the y axis runs from -8 to 8. The graph is of an increasing straight line function labeled “f”. The function starts at the point (0, 1) and increases in straight line until the point (4, 6). After this point, the function continues to increase, but at a slower rate than before, as it approaches the point (8, 8). The function does not have an x intercept and the y intercept is (0, 1).

For the following exercises (9-12), use composition to determine which pairs of functions are inverses.

  1. [latex]f(x)=8x, \,\,\, g(x)=\dfrac{x}{8}[/latex]
  2. [latex]f(x)=5x-7,\,\,\, g(x)=\dfrac{x+5}{7}[/latex]
  3. [latex]f(x)=\dfrac{1}{x-1}, \, x \ne 1,\,\,\, g(x)=\dfrac{1}{x}+1, \, x \ne 0[/latex]
  4. [latex]f(x)=x^2+2x+1, \, x \ge -1,\,\,\, g(x)=-1+\sqrt{x}, \, x \ge 0[/latex]

For the following exercises (13-17), evaluate the functions. Give the exact value.

  1. [latex]\tan^{-1}\left(\frac{\sqrt{3}}{3}\right)[/latex]
  2. [latex]\cot^{-1}(1)[/latex]
  3. [latex]\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)[/latex]
  4. [latex]\sin (\cos^{-1}\left(\frac{\sqrt{2}}{2})\right)[/latex]
  5. [latex]\tan^{-1}\left( \tan \left(-\frac{\pi}{6}\right)\right)[/latex]

For the following exercises (18-23), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.

  1. The velocity [latex]V[/latex] (in centimeters per second) of blood in an artery at a distance [latex]x[/latex] cm from the center of the artery can be modeled by the function [latex]V=f(x)=500(0.04-x^2)[/latex] for [latex]0 \le x \le 0.2[/latex].
    1. Find [latex]x=f^{-1}(V)[/latex].
    2. Interpret what the inverse function is used for.
    3. Find the distance from the center of an artery with a velocity of [latex]15[/latex] cm/sec, [latex]10[/latex] cm/sec, and [latex]5[/latex] cm/sec.
  2. The cost to remove a toxin from a lake is modeled by the function

    [latex]C(p)=\dfrac{75p}{(85-p)}[/latex],

    where [latex]C[/latex] is the cost (in thousands of dollars) and [latex]p[/latex] is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than [latex]85[/latex] ppb.

    1. Find the cost to remove [latex]25[/latex] ppb, [latex]40[/latex] ppb, and [latex]50[/latex] ppb of the toxin from the lake.
    2. Find the inverse function.
    3. Use part (b) to determine how much of the toxin is removed for [latex]$50,000[/latex].
  3. An airplane’s Mach number [latex]M[/latex] is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by [latex]\mu =2\sin^{-1}\left(\frac{1}{M}\right)[/latex].

    Find the Mach angle (to the nearest degree) for the following Mach numbers.

    An image of a birds eye view of an airplane. Directly in front of the airplane is a sideways “V” shape, with the airplane flying directly into the opening of the “V” shape. The “V” shape is labeled “mach wave”. There are two arrows with labels. The first arrow points from the nose of the airplane to the corner of the “V” shape. This arrow has the label “velocity = v”. The second arrow points diagonally from the nose of the airplane to the edge of the upper portion of the “V” shape. This arrow has the label “speed of sound = a”. Between these two arrows is an angle labeled “Mach angle”. There is also text in the image that reads “mach = M > 1.0”.

    1. [latex]M =1.4[/latex]
    2. [latex]M =2.8[/latex]
    3. [latex]M =4.3[/latex]
  4. The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function

    [latex]T(x)=5+18 \sin\left[\frac{\pi}{6}(x-4.6)\right][/latex],

    where [latex]x[/latex] is time in months and [latex]x=1.00[/latex] corresponds to January 1. Determine the month and day when the temperature is [latex]21^{\circ}[/latex] C.

  5. An object moving in simple harmonic motion is modeled by the function

    [latex]s(t)=-6 \cos \left(\frac{\pi t}{2}\right)[/latex],

    where [latex]s[/latex] is measured in inches and [latex]t[/latex] is measured in seconds. Determine the first time when the distance moved is [latex]4.5[/latex] in.

Exponential and Logarithmic Functions

For the following exercises (1-2), evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.

  1. [latex]f(x)=5^x[/latex]
    1. [latex]x=3[/latex]
    2. [latex]x=\dfrac{1}{2}[/latex]
    3. [latex]x=\sqrt{2}[/latex]
  2. [latex]f(x)=10^x[/latex]
    1. [latex]x=-2[/latex]
    2. [latex]x=4[/latex]
    3. [latex]x=\dfrac{5}{3}[/latex]

For the following exercises (3-5), match the exponential equation to the correct graph.

  1. [latex]y=3^{x-1}[/latex]
  2. [latex]y=\left(\dfrac{1}{2}\right)^x+2[/latex]
  3. [latex]y=−3^{−x}[/latex]
  1. An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -2 to 8. The graph is of a decreasing curved function. The function decreases until it approaches the line “y = 2”, but never touches this line. The y intercept is at the point (0, 3) and there is no x intercept.
  2. An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, (1/3)). Another point of the graph is at (1, 1).
  3. An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that increases until it comes close the x axis without touching it. There is no x intercept and the y intercept is at the point (0, -1). Another point of the graph is at (-1, -3).

For the following exercises (6-9), sketch the graph of the exponential function. Determine the (a) domain, (b) range, and (c) horizontal asymptote.

  1. [latex]f(x)=e^x+2[/latex]
  2. [latex]f(x)=3^{x+1}[/latex]
  3. [latex]f(x)=1-2^{−x}[/latex]
  4. [latex]f(x)=e^{−x}-1[/latex]

For the following exercises (10-13), write the equation in equivalent exponential form.

  1. [latex]\log_8 2=\dfrac{1}{3}[/latex]
  2. [latex]\log_5 25=2[/latex]
  3. [latex]\ln\left(\dfrac{1}{e^3}\right)=-3[/latex]
  4. [latex]\ln 1=0[/latex]

For the following exercises (14-18), write the equation in equivalent logarithmic form.

  1. [latex]4^{-2}=\dfrac{1}{16}[/latex]
  2. [latex]9^0=1[/latex]
  3. [latex]\sqrt[3]{64}=4[/latex]
  4. [latex]9^y=150[/latex]
  5. [latex]4^{-3/2}=0.125[/latex]

For the following exercises (19-21, sketch the graphs of the logarithmic functions and determine their

  1. domain
  2. range
  3. vertical asymptote.
  1. [latex]f(x)=\ln(x-1)[/latex]
  2. [latex]f(x)=1-\ln x[/latex]
  3. [latex]f(x)=\ln(x+1)[/latex]

For the following exercises (22-24), use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.

  1. [latex]\log_3 \dfrac{9a^3}{b}[/latex]
  2. [latex]\log_5 \sqrt{125xy^3}[/latex]
  3. [latex]\ln\left(\dfrac{6}{\sqrt{e^3}}\right)[/latex]

For the following exercises (25-28), solve the exponential equation exactly.

  1. [latex]e^{3x}-15=0[/latex]
  2. [latex]4^{x+1}-32=0[/latex]
  3. [latex]10^x=7.21[/latex]
  4. [latex]7^{3x-2}=11[/latex]

For the following exercises (29-32), solve the logarithmic equation exactly, if possible.

  1. [latex]\log_5 x=-2[/latex]
  2. [latex]\log(2x-7)=0[/latex]
  3. [latex]\log_6 (x+9)+\log_6 x=2[/latex]
  4. [latex]\ln x+\ln (x-2)=\ln 4[/latex]

For the following exercises (33-35), use the change-of-base formula and either base 10 or base [latex]e[/latex] to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

  1. [latex]\log_7 82[/latex]
  2. [latex]\log_{0.5} 211[/latex]
  3. [latex]\log_{0.2} 0.452[/latex]

For the following exercises (36-41), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.

  1. The number of bacteria [latex]N[/latex] in a culture after [latex]t[/latex] days can be modeled by the function [latex]N(t)=1300·2^{t/4}[/latex]. Find the number of bacteria present after [latex]15[/latex] days.
  2. The accumulated amount [latex]A[/latex] of a [latex]$100,000[/latex] investment whose interest compounds continuously for [latex]t[/latex] years is given by [latex]A(t)=100,000·e^{0.055t}[/latex]. Find the amount [latex]A[/latex] accumulated in [latex]5[/latex] years.
  3. According to the World Bank, at the end of 2013 ([latex]t=0[/latex]) the U.S. population was [latex]316[/latex] million and was increasing according to the following model:

    [latex]P(t)=316e^{0.0074t}[/latex],

    where [latex]P[/latex] is measured in millions of people and [latex]t[/latex] is measured in years after 2013.

    1. Based on this model, what will be the population of the United States in 2020?
    2. Determine when the U.S. population will be twice what it is in 2013.
  4. A bacterial colony grown in a lab is known to double in number in [latex]12[/latex] hours. Suppose, initially, there are [latex]1000[/latex] bacteria present.
    1. Use the exponential function [latex]Q=Q_0e^{kt}[/latex] to determine the value [latex]k[/latex], which is the growth rate of the bacteria. Round to four decimal places.
    2. Determine approximately how long it takes for [latex]200,000[/latex] bacteria to grow.
  5. The 1906 earthquake in San Francisco had a magnitude of [latex]8.3[/latex] on the Richter scale. At the same time, in Japan, an earthquake with magnitude [latex]4.9[/latex] caused only minor damage. Approximately how much more energy was released by the San Francisco earthquake than by the Japanese earthquake?