SECTION 8.4 TRIGONOMETRY

The word trigonometry comes from two greek terms, trigon, meaning triangle, and metron, meaning measure. a trigonometric ratio is a ratio of the lengths of two sides of a right triangle. By AA Similarity, a right triangle with a given acute angle is similar to every other right triangle with the same acute angle measure. So trigonometric ratios are constant for a given angle measure.

Example 1: a) Express sin L as a fraction and as a decimal to the nearest hundredth.

Example 1: b) Express cos L as a fraction and as a decimal to the nearest hundredth.

Example 1: c) Express tan L as a fraction and as a decimal to the nearest hundredth.

Example 1: d) Express sin N as a fraction and as a decimal to the nearest hundredth.

Example 1: e) Express cos N as a fraction and as a decimal to the nearest hundredth.

Example 1: f) Express tan N as a fraction and as a decimal to the nearest hundredth.

Example 2: a) Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth. Special right triangles can be used to find the sine, cosine, and tangent of 30°, 45° and 60° angles.

Example 2: b) Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth.

Example 3: 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.

Example 4: The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch? Let y be the height of the ramp from the floor in feet. The length of the ramp is 15 feet. y

If you know the sine, cosine, or tangent of an acute angle, you can use a calculator to find the measure of the angle, which is the inverse of the trigonometric ratio.

Example 5: a) Use a calculator to find the measure of  P to the nearest tenth. The measures given are those of the leg adjacent to  P and the hypotenuse, so write the equation using the cosine ratio.

Example 5: b) Use a calculator to find the measure of  D to the nearest tenth. The measures given are those of the leg opposite to  D and the hypotenuse, so write the equation using the sine ratio.

Example 6: Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree. a)

Find m  B using complementary angles. m  B ≈ 60 ° Subtract 30 from each side. So, the measure of  B is about 60 . 30 ° + m  B ≈ 90 ° m  A ≈ 30 m  A + m  B = 90° Definition of complementary angles Find AB by using the Pythagorean Theorem. (AC) 2 + (BC) 2 = (AB) 2 Pythagorean Theorem = (AB) 2 Substitution 65 = (AB) 2 Simplify. Take the positive square root of each side ≈ ABUse a calculator.

Example 6: Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree. b)

Find m  B using complementary angles. m  B ≈ 36 ° Subtract 54 from each side. So, the measure of  B is about 36 . 54 ° + m  B ≈ 90 ° m  A ≈ 54 m  A + m  B = 90° Definition of complementary angles Find AB by using the Pythagorean Theorem. (AC) 2 + (BC) 2 = (AB) 2 Pythagorean Theorem = (AB) 2 Substitution 185 = (AB) 2 Simplify. Take the positive square root of each side ≈ ABUse a calculator.