Trigonometry Ratios

Find the tangent, sine and cosine ratios of acute angles.

Use tangents, sines and cosines to find side lengths in right triangles and to solve real-

world problems.

Unit J 2

Similar Right Triangles

Every right triangle with a given acute angle must be similar.

How do we know this is true?

The AA Similarity Postulate. Each of these three triangles have two congruent angles which makes them similar.

32°

32°

32°

Unit J 3

Trigonometric Ratios

When two right triangles are similar, then the ratios of their sides are the same.

In a right triangle, these ratios are called trigonometric ratios.

Let’s see a demonstration of these ratios. They are the same for each acute angle.

Similar Right Triangles

Unit J 4

The Trigonometric Ratios

Trigonometric ratios are ratios of the sides concerning one acute angle in a right triangle.

In this triangle, let’s talk about the ratios for angle A.

The side BC is the opposite side (the side directly across from angle A).

A

B

C

AC is the adjacent side

AB is the hypotenuse

Unit J 5

The Sine Ratio

The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse.

A

B

C

sin A = h

o

y

pp

po

osit

tenu

..

s

e leg

e c

a

c a

b

Unit J 6

The Cosine Ratio

The cosine of an angle is the ratio of the length of the leg adjacent the angle to the length of the hypotenuse.

A

B

C

cos A = h

a

y

dj

po

acen

tenu

..

s

t leg

e c

b

c a

b

Unit J 7

The Tangent Ratio

The tangent of an angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle.

A

B

C

tan A = adja

oppo

cent

site le

le

..

..

g

g b

a

c a

b

Unit J 8

Get Organized: Memory Device

Memory Device for Trigonometric Functions

sin = S =

cos = C =

tan = T =

opp.hyp.

adj.hyp.

opp.adj.

OH

AH

OA

S O H

T O A

C A H

Unit J 9

Using your graphing calculator

You will be using your calculator to find the sine, cosine, or tangent functions.

Before you start, you need to make sure that your calculator is set on degrees and not radians. All angles we will be using here are to be in degrees.

First push and then make sure that the “degree” is highlighted.

To check your calculator, clear screen, push

and then “30)”. You should get 0.5SIN

MODE

Unit J 10

How do we use this information?

We can use these ratios to find missing measures in a right triangle.

xtan40

10

A

B

C

x

1040°

x0.8391

10

10 0.8391 x

8.4 x

Unit J 11

Examples of Using Trigonometric Ratios

To find x, use the sine ratio. To find y, use the cosine ratio.

A

BC63°

x

y

25

xsin63

25 25 0.8910 x

22.3 x

ycos63

25

11.3 y

25 0.45399 y