10-3 Trigonometry

Do Now

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Lesson Presentation

10-3 Trigonometry

DO NOW #14

Write each fraction as a decimal rounded to the nearest hundredth.

1. 2.

Solve each equation.

3. 4.

0.67 0.29

x = 7.25

x = 7.99

10-3 Trigonometry

SWBAT

Find the sine, cosine, and tangent of an acute angle.

Use trigonometric ratios to find side lengths in right triangles and to solve real-world problems.

By the end of today’s lesson,

Connect to Mathematical Ideas (1)(F)

10-3 Trigonometry

trigonometric ratio

sine

cosine

tangent

Vocabulary

10-3 Trigonometry

By the AA Similarity Postulate, a right triangle with

a given acute angle is similar to every other right triangle with

that same acute angle measure. So ∆ABC ~ ∆DEF ~ ∆XYZ,

and . These are trigonometric ratios. A

trigonometric ratio is a ratio of two sides of a right triangle.

𝐵𝐶

𝐴𝐶=𝐸𝐹

𝐷𝐹=𝑌𝑍

𝑋𝑍

10-3 Trigonometry

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

10-3 Trigonometry

10-3 Trigonometry

Example 1: Writing Trigonometric Ratios

What are the sine, cosine, and tangent ratios for T ?

𝐬𝐢𝐧𝑻 =opposite

hypotenuse=

8

17

𝐜𝐨𝐬𝑻 =adjacent

hypotenuse=15

17

𝐭𝐚𝐧𝑻 =opposite

adjacent=

8

15

10-3 Trigonometry

Example 2: Using a Trigonometric Ratios to Find Distance

Landmarks In 1990, the Leaning Tower of Pisa was closed to the public due to safety concerns. The tower reopened in 2001 after a 10-year project to reduce its tilt from vertical. Engineers’ efforts were successful and resulted in a tilt of 5o, reduced from 5.5o. Suppose someone drops an object from the tower at a height of 150 ft. How far from the base of the tower will the object land? Round to the nearest foot.

The given side is adjacent to the given angle. The side you want to find its opposite the given angle.

Use the tangent ratio.𝐭𝐚𝐧 5o =𝑥

150

Multiply each side by 150.𝑥 = 150(𝐭𝐚𝐧5o)

Use a calculator.𝑥 ≈13.1232995313 ft

10-3 Trigonometry

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

Caution!

10-3 Trigonometry

Example 3: Using Inverses

What is the mX in the nearest degree?

You know the lengths of the hypotenuse and the side opposite X.

You know the lengths of the hypotenuse and the side adjacent to X.

𝑼𝒔𝒆 𝒕𝒉𝒆 𝒔𝒊𝒏𝒆 𝒓𝒂𝒕𝒊𝒐. 𝑼𝒔𝒆 𝒕𝒉𝒆 𝒄𝒐𝒔𝒊𝒏𝒆 𝒓𝒂𝒕𝒊𝒐.

Write the ratio.𝐬𝐢𝐧𝑿 =6

10𝐜𝐨𝐬𝑿 =

15

20

Use the inverse.𝑿 = sin−16

10𝑿 = cos−1

15

20𝒎𝑿 ≈ 37o 𝒎𝑿 ≈ 41oUse a calculator.

10-3 Trigonometry

Example 4: Using the Tangent Ratio

∆DEG is a right triangle. You know the lengths of the side adjacent to and the side opposite DEG.

𝑼𝒔𝒆 𝒕𝒉𝒆 𝒕𝒂𝒏𝒈𝒆𝒏𝒕 𝒓𝒂𝒕𝒊𝒐.

Write the ratio.𝐭𝐚𝐧 𝐷𝐸𝐺 =150

115

Use the inverse.DEG = tan−1150

115

𝒎𝑫𝑬𝑮 ≈ 53o Use a calculator.

An urban planner needs to know the measure of the angle formed by a street and the edge of a bike path. What is the measure of DEGto the nearest degree?

10-3 Trigonometry

Got It ? Solve With Your Partner

Problem 1 Writing Trigonometric Ratios

What are the sine, cosine, and tangent ratios for G ?

𝐬𝐢𝐧𝑮 =opposite

hypotenuse=15

17

𝐜𝐨𝐬𝑮 =adjacent

hypotenuse=

8

17

𝐭𝐚𝐧𝑮 =opposite

adjacent=15

8

10-3 Trigonometry

Got It ? Solve With Your Partner

Problem 2 Using Trigonometric Ratio to Find Distance

Find the value of w to the nearest tenth.

A. B. C.

13.8 1.9 3.8

10-3 Trigonometry

Got It ? Solve With Your Partner

Problem 3 Using Trigonometric Ratio to Find Distance

A section of Filbert Street in San Francisco rises at an angle of about 17o. If you walk 150 ft up this section, what is your vertical rise? Round to the nearest foot.

44 ft

10-3 Trigonometry

Closure: Communicate Mathematical Ideas (1)(G)

How could you determine the value of sin 35o without using a calculator ?

Draw a right triangle with one acute of 35o. Measure the length of the opposite side and the hypotenuse. Then find the ratio of the length of the opposite side to the length of the hypotenuse.

10-3 Trigonometry

Use a special right triangle to write each trigonometric ratio as a fraction.

1. sin 60° 2. cos 45°

Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.

3. tan 84° 4. cos 13°

Exit Ticket: Apply Mathematics (1)(A)

Find each length. Round to the nearest tenth.

5. CB

6. AC