This document discusses the tangent ratio and how to use it to solve problems involving right triangles. It provides examples of finding the tangent of various angles, using the tangent ratio to find missing side lengths of triangles, and calculating the angle of a road given its grade. Key terms introduced are tangent ratio, sine, cosine, acute angle, opposite leg, and adjacent leg. Examples are provided to illustrate using the tangent ratio to solve problems.

1. 8-5 The Tangent Ratio

2.  Greek for “Triangle Measurement”
 You will need to use a scientific calculator to
solve some of the problems. (You can find
one online if you don’t have one.)
 Your calculator needs to be in “degree”
mode when we do trigonometry.

3.  Right triangle trigonometry
 Three ratios:
 Tangent of an angle (tanA)
 Sine of an angle (sinA)
 Cosine of an angle (cosA)
 Tangent of A (one of the acute angles)
A
B
C
opposite leg
adjacent leg

4. leg opposite Atangent of A =
leg adjacent to A
A
B
C
adjacent leg
opposite leg

5. 1. Find tan X
2. Find tan Y
1. Find tan 30°
2. Find tan 60°
Y
30°
6
X
Z
815
17

6. 1. Find tan X
2. Find tan Y
1. Find tan 30°
2. Find tan 60°
Y
30°
6
X
Z
815
17
tan 𝑌 =
15
8
tan 𝑋 =
8
15
tan 60 =
6 3
6
= 3
tan 30 =
6
6 3
=
3
3

7.  Find the value of y to the nearest tenth.
56
32
y
tan 56 =
𝑦
32
𝑦 = 32 ∗ tan 56
𝑦 ≈ 32 ∗ 1.4826
𝑦 ≈ 47.4

8. The grade of a road is the ratio of its rise to its
run and is usually given as a decimal or
percent. Find the angle that the road makes
with the horizontal if its grade is 4%.
rise
run
x
grade =
rise
run
tan x = 0.0400