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Vocabulary 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.

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Concept

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

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

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

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

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

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

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

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

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

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

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

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

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Example 2 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.

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

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A.A B.B C.C D.D 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.

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Example 3 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.

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

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

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Concept

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

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Example 4 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] nd( ÷)ENTER Answer: So, the measure of P is approximately 46.8°.

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A.A B.B C.C D.D Example 4 A.44.1° B.48.3° C.55.4° D.57.2° Use a calculator to find the measure of D to the nearest tenth.

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

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

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

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Example 5 Solve a Right Triangle Step 3Find 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. 8.06≈ ABUse a calculator.

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

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A.A B.B C.C D.D Example 5 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 Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

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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

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