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Understanding the Tangent Function: Calculator Applications & Graph Analysis

Explore the tangent function through calculator calculations, trigonometric ratios, period analysis, and graph plotting. Understand the tangent values for various angles and learn to sketch the curve effectively.

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Understanding the Tangent Function: Calculator Applications & Graph Analysis

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  1. 13.6 – The Tangent Function

  2. 3 5 5 2 6 The Tangent Function Use a calculator to find the sine and cosine of each value of . Then calculate the ratio . 1. radians 2. 30 degrees 3. 90 degrees 4. radians 5. radians 6. 0 degrees sin cos

  3. sin 3 3 cos 0.5 0.866 3 3 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 Solutions 0.866 0.5 sin 30° cos 30° 1 0 sin 90° cos 90°

  4. 2 6 6 2 sin sin cos cos 5 5 5 5 5 5 5 5 6 2 2 6 0.5 –0.866 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 Solutions (continued) 1 0 0 1 sin 0° cos 0°

  5. 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

  6. 2 period = Use the formula for the period. 1 2 = = 2 Substitute for b and simplify. One cycle occurs in the interval – to . b Asymptotes occur every 2 units, at = – , , and 3 . 1 2 Sketch the asymptotes. Plot three points in each cycle. Sketch the curve. The Tangent Function Sketch two cycles of the graph y = tan .

  7. 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. The Tangent Function What is the height of the triangle, in the design from Example 3, when = 18°? What is the height when = 20°?

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