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
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

4. 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°

5. 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

6. 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

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 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 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

10. 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°

11. 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

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

13. Extending the Trigonometric Ratios
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)