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Section 10.1 Tangent Ratios

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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

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**Tangent Ratio Examples**

Find the tan θ. A D θ θ adj hyp hyp adj. 5 B C E F opp opp. 3.6 tan θ = opp./adj. tan θ = opp./adj. tan θ = 12/5 ≈ tan θ = 3.6/2.7 ≈ 1.333

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**Finding Angles Using Tangent Ratios**

Find the indicated angle. X W R T 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°

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**Finding Side Measurements Using Tangent Ratios**

Find the indicated side. M N B 75° x x 37° D H G 18 tan 37 = x/18 tan 75 = x/12 18tan37 = x 12tan75 = x 13.56 ≈ x ≈ x

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**Finding Side Measurements Using Tangent Ratios**

Find the indicated side. M N B 53° x 42° D H G x tan 42 = 5/x tan 53 = 22/x 5/tan42 = x 22/tan53 = x 5.55 ≈ x ≈ x

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Section 10.2 Sines and Cosines

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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 cos θ, is the ratio of the length of the leg adjacent to A to the length of the hypotenuse, or opp. cos θ = adjacent/hypotenuse adj θ° hyp.

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**Sine and Cosine Ratio Examples**

Find the sin θ and cos θ. A D θ θ adj hyp hyp adj. 5 B C E F opp opp. 3.6 sin θ = opp./hyp. cos θ = adj./hyp. sin θ = opp./hyp. cos θ = adj./hyp. sin θ = 12/ cos θ = 5/ sin θ = 3.6/ cos θ = 2.7/4.5 sin θ ≈ cos θ ≈ sin θ ≈ cos θ ≈ 0.6

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**Finding Angles Using Sine and Cosine**

Find the indicated angle. X W R T Y P Find / Y. Find / W. sin Y = 6/10 cos Y = 8/10 sin W = 22.57/ 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°

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**Finding Side Measurements Using Tangent Ratios**

Find the indicated side. M N B 75° x 34° x D H G sin 34 = x/25 cos 75 = x/45 25sin34 = x 45cos75 = x 13.98 ≈ x ≈ x

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**Two Trigonometric Identities**

tan θ = sin θ/cos θ (sin θ)² + (cos θ)² = 1

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**Extending the Trigonometric Ratios**

Section 10.3 Extending the Trigonometric Ratios

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**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.

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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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∆ABC has three angles… › ∡C is a right angle › ∡A and ∡B are acute angles We can make ratios related to the acute angles in ∆ABC A CB 3 4 5.

∆ABC has three angles… › ∡C is a right angle › ∡A and ∡B are acute angles We can make ratios related to the acute angles in ∆ABC A CB 3 4 5.

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