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You will learn to:

Find trigonometric ratios using right triangles.

Solve problems using trigonometric ratios.

7-4 Objectives

Trigonometry

Trigonometric Ratios

Sine

Cosine

Tangent

Vocabulary

The word trigonometry comes from two Greek terms,

trigon, meaning triangle, and metron, meaning

measure. The study of trigonometry involves triangle

measurement. A ratio of the lengths of sides of a right

triangle is called a trigonometric ratio. The three most

common trigonometric ratios are sine, cosine, and

tangent.

Trigonometric Ratio

In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example, the sine of A is written as sin A.

Writing Math

S O H C A H T O A

Sin = Cos = Tan = Opposite

Hypotenuse

Adjacent

Hypotenuse

Opposite

Adjacent

Sine – Cosine - Tangent

Find sin L, cos L, tan L, sin N, cos N, and tan N.Express each ratio as a fraction and as a decimal.

Answer:

Find sin A, cos A, tan A, sin B, cos B, and tan B.Express each ratio as a fraction and as a decimal.

Use a calculator to find tan 56to the nearest ten thousandth.

KEYSTROKES: 56 1.482560969TAN ENTER

Answer:

KEYSTROKES: 90 0COS ENTER

Answer:

Use a calculator to find cos to the nearest ten thousandth.

Be sure your calculator is in degree mode, not radian mode.

Caution!

a. Use a calculator to find sin 48 to the nearest ten thousandth.

b. Use a calculator to find cos 85° to the nearest ten thousandth.

Answer:

Answer:

KEYSTROKES: 60 7 7.312160604SIN ENTER

Multiply each side by 60.

Use a calculator to find y.

Answer: The treadmill is about 7.3 inches high.

EXERCISING A fitness trainer sets the incline on a treadmill to 7 . The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor?

Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches.

CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about how high does the ramp rise off the ground to the nearest inch?

Answer: about 15 in.

If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle.

Inverse Trigonometric Functions

Example 2: Calculating Angle Measures from

Trigonometric Ratios

Use your calculator to find each angle measure to the nearest degree.

A. cos-1(0.87) B. sin-1(0.85) C. tan-1(0.71)

cos-1(0.87) 30° sin-1(0.85) 58° tan-1(0.71) 35°

Do not round until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator.

Caution!

Rounding

KEYSTROKES: 2 32ND ENTER)(

M x = 41.8103149

M x = 41.83594348

KEYSTROKES: 0.6672ND ENTER

Example 5: Travel Application

A highway sign warns that a section of road ahead has a 7% grade. To the nearest degree, what angle does the road make with a horizontal line?

Change the percent grade to a fraction.

A 7% grade means the road rises (or falls) 7 ft for every 100 ft of horizontal distance.

Draw a right triangle to

represent the road.

A is the angle the road

makes with a horizontal line.

Check It Out! Example 5

Baldwin St. in Dunedin, New Zealand, is the steepest street in the world. It has a grade of 38%. To the nearest degree, what angle does Baldwin St. make with a horizontal line?

Change the percent

grade to a fraction.

A 38% grade means the road rises (or falls) 38 ft for every 100 ft of horizontal distance.

Draw a right triangle to

represent the road.

A is the angle the road

makes with a horizontal line.

100 ft

38 ft

A B

C

What did you learn today?

How to:

Find trigonometric ratios using right triangles.

Solve problems using trigonometric ratios.

Assignment:

Page 368

5 – 49 odd, 59