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13.6 – The Tangent Function

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The Tangent Function Use a calculator to find the sine and cosine of each value of . Then calculate the ratio. 1. radians2.30 degrees 3.90 degrees4. radians 5. radians6.0 degrees sin cos 3 5 2 5 6

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The Tangent Function 1.Sin 0.866; cos = 0.5; 1.73 2.sin 30° = 0.5; cos 30° 0.866; 0.58 3.sin 90° = 1; cos 90° = 0; =, undefined 33 sin 3 cos 3 0.866 0.5 sin 30° cos 30° 0.5 0.866 sin 90° cos 90° 1010 Solutions

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The Tangent Function 4.sin = 0.5; cos –0.866; –0.58 5.sin = 1; cos = 0; =, undefined 6.sin 0° = 0; cos 0° = 1; = = 0 5 6 5 6 sin 5 6 cos 5 6 0.5 –0.866 5 2 5 2 sin 5 2 cos 5 2 1010 sin 0° cos 0° 0101 Solutions (continued)

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The Tangent Function Use the graph of y = tan to find each value. a. tan –45°tan –45° = –1 b. tan 0° tan 0° = 0 c. tan 45°tan 45° = 1

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Sketch the asymptotes. The Tangent Function Sketch two cycles of the graph y = tan. period = Use the formula for the period. b = = 2 Substitute for b and simplify. 1212 1212 One cycle occurs in the interval – to. Plot three points in each cycle. Sketch the curve. Asymptotes occur every 2 units, at = –,, and 3. 2

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The Tangent Function What is the height of the triangle, in the design from Example 3, when = 18°? What is the height when = 20°? Step 1: Sketch the graph.Step 2: Use the TABLE feature. When = 18°, the height of the triangle is about 32.5 ft. When = 20°, the height of the triangle is about 36.4 ft.

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