Objectives 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.

Vocabulary trigonometric ratio sine cosine tangent

_________________– a ratio of two sides of a right triangle.

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

Example 1A: Finding Trigonometric Ratios Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. sin J cos J tan K

Example 1b Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. cos A tan B sin B

Example 2a Use a special right triangle to write cos 30° as a fraction. Draw and label a 30º-60º-90º ∆.

Example 2b Use a special right triangle to write tan 45° as a fraction. Draw and label a 45º-45º-90º ∆.

Example 3A: Calculating Trigonometric Ratios Use your calculator to find the trigonometric ratio. Round to the nearest hundredth. sin 52° Be sure your calculator is in degree mode, not radian mode. Caution!

Example 3 Continued Use your calculator to find the trigonometric ratio. Round to the nearest hundredth. 3b. cos 19° 3c. tan 65° 3d. sin 62°

Example 4A: Using Trigonometric Ratios to Find Lengths Find the length. Round to the nearest hundredth. BC

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

Example 4B: Using Trigonometric Ratios to Find Lengths Find the length. Round to the nearest hundredth. QR

Example 4C: Using Trigonometric Ratios to Find Lengths Find the length. Round to the nearest hundredth. FD

Example 4d Find the length. Round to the nearest hundredth. JL

Example 5a: Problem-Solving Application The Pilatusbahn in Switzerland is the world’s steepest cog railway. Its steepest section makes an angle of about 25.6º with the horizontal and rises about 0.9 km. To the nearest hundredth of a kilometer, how long is this section of the railway track?

Understand the Problem Example 5a Continued 1 Understand the Problem Make a sketch. 0.9 km

Example 5a Continued 2 Make a Plan Solve 3

Example 5b Find AC, the length of the ramp, to the nearest hundredth of a foot.

Understand the Problem Example 5b Continued 1 Understand the Problem 2 Make a Plan

Example 5b Continued Solve 3