2 Tangent RatiosFor 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, ortan θ = opposite/adjacent
3 Tangent Ratio Examples Find the tan θ.A Dθ θadj hyp hyp adj.5B C E Fopp opp. 3.6tan θ = opp./adj. tan θ = opp./adj.tan θ = 12/5 ≈ tan θ = 3.6/2.7 ≈ 1.333
4 Finding Angles Using Tangent Ratios Find the indicated angle.X W RT Y PFind / 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 B75°x x37°D H G18tan 37 = x/18 tan 75 = x/1218tan37 = x 12tan75 = x13.56 ≈ x ≈ x
6 Finding Side Measurements Using Tangent Ratios Find the indicated side.M N B53°x42°D H Gxtan 42 = 5/x tan 53 = 22/x5/tan42 = x 22/tan53 = x5.55 ≈ x ≈ x
8 Sine and Cosine RatiosFor 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, orsin θ = opposite/hypotenuseThe 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.
9 Sine and Cosine Ratio Examples Find the sin θ and cos θ.A Dθ θadj hyp hyp adj.5B C E Fopp opp. 3.6sin 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 = 2.7/4.5sin θ ≈ cos θ ≈ sin θ ≈ cos θ ≈ 0.6
10 Finding Angles Using Sine and Cosine Find the indicated angle.X W RT Y PFind / 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°
11 Finding Side Measurements Using Tangent Ratios Find the indicated side.M N B75°x34° xD H Gsin 34 = x/25 cos 75 = x/4525sin34 = x 45cos75 = x13.98 ≈ x ≈ x
12 Two Trigonometric Identities tan θ = sin θ/cos θ (sin θ)² + (cos θ)² = 1
13 Extending the Trigonometric Ratios Section 10.3Extending 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 CircleThe 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)