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Chapter 7

Trigonometry Basics

Trigonometric functions are special in several ways. The first characteristic that separates them from all the other types of functions is that input values are always angle measures. You input an angle measure, and the output is some real number. The angle measures can be in degrees or radians — a degree being one-360th of a slice of a circle, and a radian being about one-sixth of a circle. Each type has its place and use in the study of trigonometry.

Another special feature of trig functions is their periodicity; the function values repeat over and over and over, infinitely. This predictability works well with many types of physical phenomena, so trigonometric functions serve as models for many naturally occurring observations.

The Problems You’ll Work On

In this chapter, you’ll work with trigonometric functions and their properties in the following ways:

 Defining the basic trig functions using the sides of a right triangle

 Expanding the input values of trig functions by using the unit circle

 Exploring the right triangle and trig functions to solve practical problems

 Working with special right triangles and their unique ratios

 Changing angle measures from degrees to radians and vice versa

 Determining arc length of pieces of circles

 Using inverse trig functions to solve for angle measures

 Solving equations involving trig functions

What to Watch Out For

Don’t let common mistakes trip you up; keep in mind that when working with trigonometric functions, some challenges will include the following:

 Setting up ratios for the basic trig functions correctly

 Recognizing the corresponding sides of right triangles when doing applications

 Remembering the counterclockwise rotation in the standard position of angles

 Measuring from the terminal side to the x-axis when determining reference angles

 Keeping the trig functions and their inverses straight from the functions and their reciprocals

Using Right Triangles to Determine Trig Functions

391−396 Use the triangle to find the trig ratio.

391.

Illustration by Thomson Digital

392.

Illustration by Thomson Digital

393.

Illustration by Thomson Digital

394.

Illustration by Thomson Digital

395.

Illustration by Thomson Digital

396.

Illustration by Thomson Digital

Solving Problems by Using Right Triangles and Their Functions

397−406 Solve the problem. Round your answer to the nearest tenth unless otherwise indicated.

397. A ladder leaning against a house makes an angle of 30° with the ground. The foot of the ladder is 7 feet from the foot of the house. How long is the ladder?

398. A child flying a kite lets out 300 feet of string, which makes an angle of 35° with the ground. Assuming that the string is straight, how high above the ground is the kite?

399. An airplane climbs at an angle of 9° with the ground. Find the ground distance it has traveled when it reaches an altitude of 400 feet.

400. A car is traveling up a grade with an angle of elevation of 4°. After traveling 1 mile, what is the vertical change in feet?

401. Jase is in a hot air balloon that is 600 feet above the ground, where they can see their brother Willie. The angle of depression from Jase’s line of sight to Willie is 25°. How far is Willie from the point on the ground directly below the hot air balloon?

402. A bird is flying at a height of 36 feet and spots a windowsill 8 feet off the ground on which to perch. If the windowsill is at a 28° angle of depression from the bird, how far must the bird fly before it can land?

403. A submarine traveling 8 mph is descending at an angle of depression of 5°. How many minutes does the submarine take to reach a depth of 90 feet?

404. You’re a block away from a building that is 820 feet tall. Your friend is between you and the building. The angle of elevation from your position to the top of the building is 42°. The angle of elevation from your friend’s position to the top of the building is 71°. To the nearest foot, how far are you from your friend?

405. A lighthouse is located at the top of a hill 100 feet tall. From a point P on the ground, the angle of elevation to the top of the hill is 25°. From the same point P, the angle of elevation to the top of the lighthouse is 50°. How tall is the lighthouse?

406. A surveyor needs to find the distance BC across a lake as part of a project to build a bridge. The distance from point A to point B is 250 feet. The measurement of angle A is 40°, and the measurement of angle B is 112°. What is the distance BC across the lake to the nearest foot?

Illustration by Thomson Digital

Working with Special Right Triangles

407−410 Find the length of the side indicated by x.

407.

Illustration by Thomson Digital

408.

Illustration by Thomson Digital

409.

Illustration by Thomson Digital

410.

Illustration by Thomson Digital

Changing Radians to Degrees

411−415 Convert the radian measure to degrees. Round any decimals to two places.

411.

412.

413.

414. 3

415. 0.93

Changing Degrees to Radians

416−420 Convert the degree measure to radians. Give the exact measure in terms of .

416. 225°

417. 60°

418. −405°

419. 36°

420. 167°

Finding Angle Measures (in Degrees) in Standard Position

421−422 The terminal side of an angle in standard position on the unit circle contains the given point. Give the degree measure of .

421.

422.

Determining Angle Measures (in Radians) in Standard Position

423−425 The terminal side of an angle in standard position on the unit circle contains the given point. Give the radian measure of .

423.

424.

425.

Identifying Reference Angles

426−430 Find the reference angle for the angle measure. (Recall that the quadrants in standard position are numbered counterclockwise, starting in the upper right-hand corner.)

426. 167°

427. 342°

428. 265°

429. 792°

430. −748°

Determining Trig Functions by Using the Unit Circle

431−434 Find the exact value of the trigonometric function. If any are not defined, write undefined.

431. sin 225°

432. tan 330°

433. cos 405°

434.

Calculating Trig Functions by Using Other Functions and Terminal Side Positions

435−440 Use the given information to find the exact value of the trigonometric function.

435. Given: ; is in quadrant I

Find:

436. Given: ; is in quadrant II

Find:

437. Given: is in quadrant IV

Find:

438. Given: ; is in quadrant III.

Find:

439. Given: ;

Find:

440. Given: ,

Find:

Using the Arc Length Formula

441−445 Solve the problem using the arc length formula , where r is the radius of the circle and is in radians.

441. The central angle in a circle with a radius of 20 cm is Find the exact length of the intercepted arc.

442. The central angle in a circle of radius 6 cm is 85°. Find the exact length of the intercepted arc.

443. Find the radian measure of the central angle that intercepts an arc with a length of inches in a circle with a radius of 13 inches.

444. The second hand of a clock is 18 inches long. In 25 seconds, it sweeps through an angle of 150°. How far does the tip of the second hand travel in 25 seconds?

445. How far does the tip of a 15 cm long minute hand on a clock move in 10 minutes?

Evaluating Inverse Functions

446−450 Find an exact value of y.

446. ; give y in radians.

447. ; give y in radians.

448. ; give y in radians.

449. ; give y in radians.

450.

Solving Trig Equations for x in Degrees

451−455 Find all solutions of the equation in the interval . (Recall that the quadrants in standard position are numbered counterclockwise, starting in the upper right-hand corner.)

451.

452.

453.

454.

455.

Calculating Trig Equations for x in Radians

456−460 Find all solutions of the equation in the interval . (Recall that the quadrants in standard position are numbered counterclockwise, starting in the upper right-hand corner.)

456.

457.

458.

459.

460.