Trigonometry--The study of the properties of triangles Trigonometry--The study of the properties of triangles. Trigonometry means angle measurement. Trigonometric Ratio--The ratios of the measures of two sides of a right triangle. Vocabulary

Concept

Find Sine, Cosine, and Tangent Ratios A. Express sin L as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

Find Sine, Cosine, and Tangent Ratios B. Express cos L as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

Find Sine, Cosine, and Tangent Ratios C. Express tan L as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

Find Sine, Cosine, and Tangent Ratios D. Express sin N as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

Find Sine, Cosine, and Tangent Ratios E. Express cos N as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

Find Sine, Cosine, and Tangent Ratios F. Express tan N as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

A. Find sin A. A. B. C. D. A B C D Example 1

B. Find cos A. A. B. C. D. A B C D Example 1

C. Find tan A. A. B. C. D. A B C D Example 1

D. Find sin B. A. B. C. D. A B C D Example 1

E. Find cos B. A. B. C. D. A B C D Example 1

F. Find tan B. A. B. C. D. A B C D Example 1

The side adjacent to the 60° angle has a measure of x. Use Special Right Triangles to Find Trigonometric Ratios Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth. Draw and label the side lengths of a 30°-60°-90° right triangle, with x as the length of the shorter leg and 2x as the length of the hypotenuse. The side adjacent to the 60° angle has a measure of x. Example 2

Definition of cosine ratio Use Special Right Triangles to Find Trigonometric Ratios Definition of cosine ratio Substitution Simplify. Example 2

Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth. A. B. C. D. A B C D Example 2

Estimate Measures Using Trigonometry 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. Example 3

Use a calculator to find y. Estimate Measures Using Trigonometry Multiply each side by 60. Use a calculator to find y. KEYSTROKES: 60 7 7.312160604 ENTER SIN Answer: The treadmill is about 7.3 inches high. Example 3

CONSTRUCTION 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? A. 1 in. B. 11 in. C. 16 in. D. 15 in. A B C D Example 3

Concept

Use a calculator to find the measure of P to the nearest tenth. Find Angle Measures Using Inverse Trigonometric Ratios Use a calculator to find the measure of P to the nearest tenth. Example 4

Answer: So, the measure of P is approximately 46.8°. Find Angle Measures Using Inverse Trigonometric Ratios The measures given are those of the leg adjacent to P and the hypotenuse, so write the equation using the cosine ratio. KEYSTROKES: [COS] 13 19 46.82644889 2nd ( ÷ ) ENTER Answer: So, the measure of P is approximately 46.8°. Example 4

Use a calculator to find the measure of D to the nearest tenth. B. 48.3° C. 55.4° D. 57.2° A B C D Example 4

Solve a Right Triangle Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree. Example 5

Step 1 Find mA by using a tangent ratio. Solve a Right Triangle Step 1 Find mA by using a tangent ratio. Definition of inverse tangent 29.7448813 ≈ mA Use a calculator. So, the measure of A is about 30. Example 5

Step 2 Find mB using complementary angles. Solve a Right Triangle Step 2 Find mB using complementary angles. mA + mB = 90 Definition of complementary angles 30 + mB ≈ 90 mA ≈ 30 mB ≈ 60 Subtract 30 from each side. So, the measure of B is about 60. Example 5

Step 3 Find AB by using the Pythagorean Theorem. Solve a Right Triangle Step 3 Find AB by using the Pythagorean Theorem. (AC)2 + (BC)2 = (AB)2 Pythagorean Theorem 72 + 42 = (AB)2 Substitution 65 = (AB)2 Simplify. Take the positive square root of each side. 8.06 ≈ AB Use a calculator. Example 5

So, the measure of AB is about 8.06. Solve a Right Triangle So, the measure of AB is about 8.06. Answer: mA ≈ 30, mB ≈ 60, AB ≈ 8.06 Example 5

Solve the right triangle Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. A. mA = 36°, mB = 54°, AB = 13.6 B. mA = 54°, mB = 36°, AB = 13.6 C. mA = 36°, mB = 54°, AB = 16.3 D. mA = 54°, mB = 36°, AB = 16.3 A B C D Example 5

Summary: If you are given the angle, use sin, cos, and tan If you want to find the angle, use sin-1, cos-1, and tan-1