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TRIGONOMETRIC RATIOS

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TRIGONOMETRIC RATIOS

goal: know how to set up different trig ratios

θ this is the symbol for an unknown angle measure.

It’s name is ‘Theta’.

Don’t let it scare you… it’s like ‘x’ except for angle measure… it’s a way for us to keep our variables understandable and organized.

Where to start…

Trigonometric Ratios

hypotenuse

hypotenuse

opposite

opposite

adjacent

adjacent

One more time…

Here are the ratios:

O

S

sinθ = opposite side

hypotenuse

H

A

C

cosθ = adjacent side

hypotenuse

H

O

T

tanθ =opposite side

adjacent side

A

SOH CAH TOA

We could ask for the trig functions of the angleΘ by using the definitions.

c

b

SOHCAHTOA

SOHCAHTOA

Θ

INE

a

ANGENT

OSINE

PPOSITE

DJACENT

PPOSITE

DJACENT

YPOTENUSE

YPOTENUSE

Oh, I'm acute!

So am I!

You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The hypotenuse will always be the longest side and will always be opposite the right angle.

This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle.

5

4

Θ

3

Let ∆ABC be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows.

Side adjacent to A

b

cos A =

=

hypotenuse

c

Side opposite A

a

sin A =

=

hypotenuse

c

Side opposite A

a

tan A =

=

Side adjacent to A

b

Let’s practice…

Write the ratio for sin L

Sin L= _a

c

Write the ratio for cosL

Cos L = _b_

c

Write the ratio for tan L

Tan L = _a_

b

M

c

a

N b L

Let’s switch angles: Find the sin, cos and tan for Angle M:

Tan M = _b_

a

Sin M = _b_

c

Cos M = _a_

c

Given each triangle, write the ratio that could be used to find x by connecting the angle and sides given.

a

32

x

b

x

65

Given the triangle, write all the ratios that could be used to find x by connecting the angle and sides given.

d

c

x

56

opposite

sin S =

hypotenuse

adjacent

cosS =

hypotenuse

opposite

tanS =

adjacent

opposite

sin S =

hypotenuse

adjacent

cosS =

hypotenuse

opposite

tanS =

adjacent

opposite

sin A =

hypotenuse

adjacent

cosA =

hypotenuse

opposite

tanA =

adjacent

opposite

sin 45=

hypotenuse

adjacent

cos 45=

hypotenuse

opposite

tan 45=

adjacent

√2

45

opposite

sin 30=

hypotenuse

adjacent

cos 30=

hypotenuse

opposite

tan 30=

adjacent

30

√3