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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

Section 10.1

Tangent Ratios

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
  • tan θ = opposite/adjacent
tangent ratio examples
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

finding angles using tangent ratios
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°

finding side measurements using tangent ratios
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

finding side measurements using tangent ratios1
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

section 10 2

Section 10.2

Sines and Cosines

sine and cosine ratios
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.
sine and cosine ratio examples
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

finding angles using sine and cosine
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°

finding side measurements using tangent ratios2
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

two trigonometric identities
Two Trigonometric Identities
  • tan θ = sin θ/cosθ (sin θ)² + (cosθ)² = 1
section 10 3

Section 10.3

Extending the Trigonometric Ratios

extending angle measure
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.
the unit circle
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)