8.5 and 8.6 Trigonometric Ratios

Agenda and Objectives

• Agenda– Finish quiz (review first before I give it

back to you)– Start notes/powerpoint on trig ratios– Classwork– Start homework

• Objective– Find the sine, the cosine, and the tangent

of an acute triangle.

Finding Trig Ratios

• A trig ratio is a ratio of the lengths of two sides of a right triangle.

• The word trigonometry is derived from the ancient Greek language and means measurement of triangles.

• The three basic trig ratios are sine, cosine, and tangent.

• Abbreviated as sin, cos, and tan respectively

Trigonometric Ratios

• Let ∆ABC be a right triangle. The sin, the cos, and the tan of the acute angle A are defined as follows:

ac

bside adjacent to angle A

Sideoppositeangle A

hypotenuse

A

B

C

sin A =Side opposite A

hypotenuse=

a

c

cos A =Side adjacent to A

hypotenuse=

b

c

tan A =Side opposite A

Side adjacent to A=

a

b

The famous Indian…

SOHCAHTOASin

Cos

Tan

Ex. 1: Find the sin, the cos, and the tan of the indicated angle.

S

sin S = opposite

hypotenuse

cosS = adjacent

hypotenuse

tanS = opposite

adjacent

5

13≈ 0.3846

12

13≈ 0.9231

5

12≈ 0.4167

adjacent

opposite

12

13 hypotenuse5

R

T S

Ex. 1: Find the sin, the cos, and the tan of the indicated angle.

R

sin R = opposite

hypotenuse

cosR= adjacent

hypotenuse

tanR = opposite

adjacent

12

13≈ 0.9231

5

13≈ 0.3846

12

5≈ 2.4

adjacent

opposite12

13 hypotenuse5

R

T S

Ex. 2: Find the sin, the cos, and the tan of 45

45

sin 45= opposite

hypotenuse

adjacent

hypotenuse

opposite

adjacent

1

hypotenuse1

√2

cos 45=

tan 45=

1

√2=

√2

2≈ 0.7071

1

√2=

√2

2≈ 0.7071

1

1= 1

You could begin by sketching the special triangle 45-45-90, but let’s open your book to page 311 and look it up!

45

2

1

You try: Find the sin, the cos, and the tan of 30

Triangle Book

sin 30= opposite

hypotenuse

adjacent

hypotenuse

opposite

adjacent

√3

cos 30=

tan 30=

First try using the triangle at the right to figure out the measure. Then compare your answers to the table on page 311.

30

Ex: 5 Using a Calculator• You can use a calculator to

approximate the sin, cos, and the tan of 74. Make sure that your calculator is in degree mode.

• The following table shows some sample keystroke sequences accepted by most calculators.

Sample keystrokes

More examples

Independent Practice

• Page 306 1-7• Page 313 1-7

• HOMEWORK– Page 308 2-18 even– page 314 2-12 even