1 Million+ Step-by-step solutions

Why do the values of cos^{-1} x lie in the interval [0, π]?

Is it true that tan (tan^{-1} x) = x for all x? Is it true that tan^{-1} (tan x) = x for all x?

Sketch the graphs of y = cos x and y = cos^{-1} x on the same set of axes.

The function tan x is undefined at x = ± π/2. How does this fact appear in the graph of y = tan^{-1} x?

State the domain and range of sec^{-1} x.

Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.

cos (2π/3)

Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.

sin (2π/3)

Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians.

tan (-3π/4)

tan (15π/4)

cot (-13π/3)

sec (7π/6)

cot (-17π/3)

sin (16π/3)

Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work.

cos 0

Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work.

sin (-π/2)

Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work.

cos (-π)

tan 3π

sec (5π/2)

cot π

Prove that sec θ = 1/cos θ.

Prove that tan θ = sin θ/cos θ.

Prove that tan^{2} θ + 1 = sec^{2} θ.

Prove that sin θ/csc θ + cos θ/sec θ = 1.

Prove that sec (π/2 - θ) = csc θ.

Prove that sec (x + p) = -sec x.

Find the exact value of cos (π/12).

Find the exact value of tan (3π/8).

Solve the following equations.

tan x = 1

Solve the following equations.

2θ cos θ + θ = 0

Solve the following equations.

sin^{2} θ = 1/4, 0 ≤ θ < 2π

Solve the following equations.

cos^{2} θ = 1/2, 0 ≤ θ < 2π

Solve the following equations.

√2 sin x - 1 = 0

Solve the following equations.

sin 3x = √2/2, 0 ≤ x < 2π

Solve the following equations.

cos 3x = sin 3x, 0 ≤ x < 2π

Solve the following equations.

sin^{2} θ - 1 = 0

Solve the following equations.

sin θ cos θ = 0, 0 ≤ θ < 2π

Solve the following equations.

tan^{2} 2θ = 1, 0 ≤ θ < π

Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

sin^{-1} 1

Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

cos^{-1} (-1)

Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

tan^{-1} 1

cos^{-1} (-√2/2)

sin^{-1} √3/2

cos^{-1} 2

cos^{-1} (-1/2)

sin^{-1} (-1)

cos (cos^{-1} (-1))

cos^{-1} (cos (7π/6))

Draw a right triangle to simplify the given expressions. Assume x > 0.

cos (sin^{-1} x)

Draw a right triangle to simplify the given expressions. Assume x > 0.

cos (sin^{-1} (x/3))

Draw a right triangle to simplify the given expressions. Assume x > 0.

sin (cos^{-1} (x/2))

Draw a right triangle to simplify the given expressions. Assume x > 0.

sin^{-1} (cos θ), for 0 ≤ θ ≤ π/2

Draw a right triangle to simplify the given expressions. Assume x > 0.

sin (2 cos^{-1} x)

Draw a right triangle to simplify the given expressions. Assume x > 0.

cos (2 sin^{-1} x)

Prove the following identities.

cos^{-1} x + cos^{-1} (-x) = π

Prove the following identities.

sin^{-1} y + sin^{-1} (-y) = 0

Sketch a graph of the given pair of functions to conjecture a relationship between the two functions. Then verify the conjecture.

sin^{-1} x; π/2 - cos^{-1} x

Sketch a graph of the given pair of functions to conjecture a relationship between the two functions. Then verify the conjecture.

tan^{-1} x; π/2 - cot^{-1} x

Without using a calculator, evaluate or simplify the following expressions.

tan^{-1} √3

Without using a calculator, evaluate or simplify the following expressions.

cot^{-1} (-1/√3)

Without using a calculator, evaluate or simplify the following expressions.

sec^{-1} 2

Without using a calculator, evaluate or simplify the following expressions.

csc^{-1} (-1)

Without using a calculator, evaluate or simplify the following expressions.

tan^{-1} (tan (π/4))

Without using a calculator, evaluate or simplify the following expressions.

tan^{-1} (tan (3π/4))

Without using a calculator, evaluate or simplify the following expressions.

csc^{-1} (sec 2)

Without using a calculator, evaluate or simplify the following expressions.

tan (tan^{-1} 1)

Use a right triangle to simplify the given expressions. Assume x > 0.

cos (tan^{-1} x)

Use a right triangle to simplify the given expressions. Assume x > 0.

tan (cos^{-1} x)

Use a right triangle to simplify the given expressions. Assume x > 0.

cos (sec^{-1} x)

Use a right triangle to simplify the given expressions. Assume x > 0.

cot (tan^{-1} 2x)

Use a right triangle to simplify the given expressions. Assume x > 0.

Use a right triangle to simplify the given expressions. Assume x > 0.

Express u in terms of x using the inverse sine, inverse tangent, and inverse secant functions.

Express u in terms of x using the inverse sine, inverse tangent, and inverse secant functions.

Determine whether the following statements are true and give an explanation or counterexample.

a. sin (a + b) = sin a + sin b.

b. The equation cos θ = 2 has multiple real solutions.

c. The equation sin θ = 1/2 has exactly one solution.

d. The function sin (πx/12) has a period of 12.

e. Of the six basic trigonometric functions, only tangent and cotangent have a range of (- ∞, ∞).

f. sin^{-1} x/cos^{-1} x = tan^{-1} x.

g. cos^{-1} (cos (15π/16)) = 15π/16.

h. sin^{-1} x = 1/sin x.

Given the following information about one trigonometric function, evaluate the other five functions.

sin θ = - 4/5 and π < θ < 3π/2

Given the following information about one trigonometric function, evaluate the other five functions.

cos θ = 5/13 and 0 < θ < π/2

Given the following information about one trigonometric function, evaluate the other five functions.

sec θ = 5/3 and 3π/2 < θ < 2π

Given the following information about one trigonometric function, evaluate the other five functions.

csc θ = 13/12 and 0 < θ < π/2

Identify the amplitude and period of the following functions.

f(θ) = 2 sin 2θ

Identify the amplitude and period of the following functions.

g(θ) = 3 cos (θ/3)

Identify the amplitude and period of the following functions.

p(t) = 2.5 sin (12/(t - 3))

Identify the amplitude and period of the following functions.

q(x) = 3.6 cos (πx/24)

Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.

f(x) = 3 sin 2x

Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.

g(x) = -2 cos (x/3)

Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.

p(x) = 3 sin (2x - π/3) + 1

q(x) = 3.6 cos (πx/24) + 2

It has a period of 12 hr with a minimum value of -4 at t = 0 hr and a maximum value of 4 at t = 6 hr.

Design a sine function with the given properties.

It has a period of 24 hr with a minimum value of 10 at t = 3 hr and a maximum value of 16 at t = 15 hr.

Design a sine function with the given properties.

Near the end of the 1950 Rose Bowl football game between the University of California and Ohio State University, Ohio State was preparing to attempt a field goal from a distance of 23 yd from the endline at point A on the edge of the kicking region (see figure). But before the kick, Ohio State committed a penalty and the ball was backed up 5 yd to point B on the edge of the kicking region. After the game, the Ohio State coach claimed that his team deliberately committed a penalty to improve the kicking angle. Given that a successful kick must go between the uprights of the goal posts G_{1} and G_{2}, is ∠G_{1}BG_{2} greater than ∠G_{1}AG_{2}? (In 1950, the uprights were 23 ft 4 in apart, equidistant from the origin on the end line. The boundaries of the kicking region are 53 ft 4 in apart and are equidistant from the y-axis.

The Earth is approximately circular in cross section, with a circumference at the equator of 24,882 miles. Suppose we use two ropes to create two concentric circles: one by wrapping a rope around the equator and another using a rope 38 ft longer (see figure). How much space is between the ropes?

Verify that the function has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset.

a. It has a period of 365 days.

b. Its maximum and minimum values are 14.8 and 9.2, respectively,which occur approximately at t = 172 and t = 355, respectively (corresponding to the solstices).

c. D(81) = 12 and D(264) ≈ 12 (corresponding to the equinoxes).

A light block hangs at rest from the end of a spring when it is pulled down 10 cm and released. Assume the block oscillates with an amplitude of 10 cm on either side of its rest position with a period of 1.5 s. Find a trigonometric function d(t) that gives the displacement of the block t seconds after it is released, where d(t) > 0 represents downward displacement.

A boat approaches a 50-ft-high lighthouse whose base is at sea level. Let d be the distance between the boat and the base of the lighthouse. Let L be the distance between the boat and the top of the lighthouse. Let θ be the angle of elevation between the boat and the top of the lighthouse.

a. Express d as a function of θ.

b. Express L as a function of θ.

Two ladders of length a lean against opposite walls of an alley with their feet touching (see figure). One ladder extends h feet up the wall and makes a 75° angle with the ground. The other ladder extends k feet up the opposite wall and makes a 45° angle with the ground. Find the width of the alley in terms of h. Assume the ground is horizontal and perpendicular to both walls.

A pole of length L is carried horizontally around a corner where a 3-ft-wide hallway meets a 4-ft-wide hallway. For 0 < θ < π/2, find the relationship between L and u at the moment when the pole simultaneously touches both walls and the corner P. Estimate u when L = 10 ft.

The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21). However, the latest sunrise and the earliest sunset do not occur on the winter solstice, and the earliest sunrise and the latest sunset do not occur on the summer solstice. At latitude 40° north, the latest sunrise occurs on January 4 at 7:25 a.m. (14 days after the solstice), and the earliest sunset occurs on December 7 at 4:37 p.m. (14 days before the solstice). Similarly, the earliest sunrise occurs on July 2 at 4:30 a.m. (14 days after the solstice) and the latest sunset occurs on June 7 at 7:32 p.m. (14 days before the solstice). Using sine functions, devise a function s(t) that gives the time of sunrise t days after January 1 and a function S(t) that gives the time of sunset t days after January 1. Assume that s and S are measured in minutes and s = 0 and S = 0 correspond to 4:00 a.m. Graph the functions. Then graph the length of the day function D(t) = S(t) - s(t) and show that the longest and shortest days occur on the solstices.

An auditorium with a flat floor has a large flatpanel television on one wall. The lower edge of the television is 3 ft above the floor, and the upper edge is 10 ft above the floor (see figure). Express θ in terms of x.

Prove that the area of a sector of a circle of radius r associated with a central angle θ (measured in radians) is A = 1/2 r^{2} θ.

Use the figure to prove the law of cosines (which is a generalization of the Pythagorean theorem): c^{2} = a^{2} + b^{2} - 2ab cos θ.

Use the figure to prove the law of sines: sin A/a = sin B/b = sin C/c.

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