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Section 10.1

Section 10.1. Tangent Ratios . Tangent Ratios. For a given acute angle / A with a measure of θ° , the tangent of / A, or tan θ , is the ratio of the length of the leg opposite / A to the length of the leg adjacent to / A in any right triangle having A as one vertex, or

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Section 10.1

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  1. Section 10.1 Tangent Ratios

  2. Tangent Ratios • For a given acute angle / A with a measure of θ°, the tangent of / A, or tan θ, is the ratio of the length of the leg opposite / A to the length of the leg adjacent to / A in any right triangle having A as one vertex, or • tan θ = opposite/adjacent

  3. Tangent Ratio Examples • Find the tan θ. A D θ134.5θ adj. hyp. hyp.2.7 adj. 5 B C E F opp. 12opp. 3.6 tan θ = opp./adj. tan θ = opp./adj. tan θ = 12/5 ≈ 2.4 tan θ = 3.6/2.7 ≈ 1.333

  4. Finding Angles Using Tangent Ratios • Find the indicated angle. X W 12 R 6 22.57 T 8 Y P Find / Y. Find / W. tan Y = 6/8 tan W = 22.57/12 / Y = tan⁻¹(6/8) / W = tan⁻¹(22.57/12) / Y = 36.87°/ W = 62°

  5. Finding Side Measurements Using Tangent Ratios • Find the indicated side. M N B 75° x 12 x 37° D H G 18 tan 37 = x/18 tan 75 = x/12 18tan37 = x 12tan75 = x 13.56 ≈ x 44.78 ≈ x

  6. Finding Side Measurements Using Tangent Ratios • Find the indicated side. M N B 53° 5 x 22 42° D H G x tan 42 = 5/x tan 53 = 22/x 5/tan42 = x 22/tan53 = x 5.55 ≈ x 16.58 ≈ x

  7. Section 10.2 Sines and Cosines

  8. Sine and Cosine Ratios • For a given angle / A with a measure of θ°, the sine of / A, or sin θ, is the ratio of the length of the leg opposite A to the length of the hypotenuse in a right triangle with A as one vertex, or • sin θ = opposite/hypotenuse • The cosine of / A, or cosθ, is the ratio of the length of the leg adjacent to A to the length of the hypotenuse, or opp. • cosθ = adjacent/hypotenuse adjθ°hyp.

  9. Sine and Cosine Ratio Examples • Find the sin θ and cosθ. A D θ134.5θ adj. hyp. hyp.2.7 adj. 5 B C E F opp. 12opp. 3.6 sin θ = opp./hyp. cosθ = adj./hyp. sin θ = opp./hyp. cosθ = adj./hyp. sin θ = 12/13 cosθ = 5/13 sin θ = 3.6/4.5 cosθ = 2.7/4.5 sin θ ≈ 0.92 cosθ ≈ 0.38 sin θ ≈ 0.8 cosθ ≈ 0.6

  10. Finding Angles Using Sine and Cosine • Find the indicated angle. X W 12 R 6 10 25.56 22.57 T 8 Y P Find / Y. Find / W. sin Y = 6/10 cos Y = 8/10 sin W = 22.57/25.56 cos W = 12/25.56 / Y = sin⁻¹(6/10) / Y = cos⁻¹(8/10) / W = sin⁻¹(22.57/25.56) / W = cos⁻¹(12/25.56) / Y ≈ 36.87° / Y ≈ 36.87°/ W = 62° / W = 62°

  11. Finding Side Measurements Using Tangent Ratios • Find the indicated side. M N 45 B 75° x 25 34° x D H G sin 34 = x/25 cos 75 = x/45 25sin34 = x 45cos75 = x 13.98 ≈ x 9.36 ≈ x

  12. Two Trigonometric Identities • tan θ = sin θ/cosθ (sin θ)² + (cosθ)² = 1

  13. Section 10.3 Extending the Trigonometric Ratios

  14. Extending Angle Measure • Imagine a ray with its endpoint at the origin of a coordinate plane and extending along the positive x-axis. Then imagine the ray rotating a certain number of degrees, say θ, counterclockwise about the origin. θ can be any number of degrees, including numbers greater than 360°. A figure formed by a rotating ray and a stationary reference ray, such as the positive x-axis, is called an angle of rotation.

  15. The Unit Circle • The unit circle is a circle with its center at the origin and a radius of 1. • In the language of transformations, it consists of all the rotation images of the point P(1, 0) about the origin. P(1, 0)

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