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.

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TRIGONOMETRIC RATIOS

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Trigonometric ratios

TRIGONOMETRIC RATIOS

goal: know how to set up different trig ratios


Trigonometric 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

Trigonometric Ratios


Trigonometric ratios

hypotenuse

hypotenuse

opposite

opposite

adjacent

adjacent


Trigonometric ratios

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


Trigonometric ratios

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


Trigonometric ratios

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


Trigonometric ratios1

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

Trigonometric Ratios

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


Trigonometric ratios

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


Practice together

Practice Together:

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


You do

YOU DO:

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


Ex 2 finding trig ratios

Ex. 2: Finding Trig Ratios

opposite

sin S =

hypotenuse

adjacent

cosS =

hypotenuse

opposite

tanS =

adjacent


Ex 2 finding trig ratios find the sine the cosine and the tangent of the indicated angle

Ex. 2: Finding Trig Ratios—Find the sine, the cosine, and the tangent of the indicated angle.

opposite

sin S =

hypotenuse

adjacent

cosS =

hypotenuse

opposite

tanS =

adjacent


Ex 1 finding trig ratios

Ex. 1: Finding Trig Ratios

opposite

sin A =

hypotenuse

adjacent

cosA =

hypotenuse

opposite

tanA =

adjacent


Ex 3 finding trig ratios find the sine the cosine and the tangent of 45

Ex. 3: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 45

opposite

sin 45=

hypotenuse

adjacent

cos 45=

hypotenuse

opposite

tan 45=

adjacent

√2

45


Ex 4 finding trig ratios find the sine the cosine and the tangent of 30

Ex. 4: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 30

opposite

sin 30=

hypotenuse

adjacent

cos 30=

hypotenuse

opposite

tan 30=

adjacent

30

√3


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