Trigonometric

Ratios

A RATIO is a comparison of two numbers. For

example; boys to girls cats : dogs

right : wrong.

In Trigonometry, the comparison is between

sides of a triangle.

We need to do some housekeeping before we

can proceed…

In trigonometry, the ratio we are talking about is the comparison of the sides of a

RIGHT TRIANGLE.

Two things MUST BE understood:1. This is the hypotenuse.. This

will ALWAYS be the hypotenuse2. This is 90°… this makes the

right triangle a right triangle…. Without it, we can not do this trig… we WILL NOT use it in our calculations because we COULD NOT do calculations without it.

Now that we agree about the hypotenuse and right angle, there are only 4 things left; the 2 other

angles and the 2 other sides.

A We will refer to the sides in terms of their proximity to the angle

If we look at angle A, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse.

opposite

adjacent

hypotenuse

B

If we look at angle B, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse.

opposite

adjacent

hypotenuse

Remember we won’t use the right angle

X

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.

One more thing…

Here we go!!!!

Trigonometric RatiosName“say”

Sine Cosine tangent

AbbreviationAbbrev.

Sin Cos Tan

Ratio of an angle measure

Sinθ = opposite side hypotenuse

cosθ = adjacent side hypotenuse

tanθ =opposite side adjacent side

One more time…Here are the ratios:

One more time…Here are the ratios:

sinθ = opposite side hypotenuse

cosθ = adjacent side hypotenuse

tanθ =opposite side adjacent side

S OH

AHOA

C

T

SOH CAH TOA

Let’s practice…

B

c

a

C b A

Write the ratio for sin A

Sin A = a c

Write the ratio for cos A

Cos A = b c

Write the ratio for tan A

Tan A = a b

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

Sin B = b

cCos B = a

c

Tan B = b

a

Make sure you have a calculator…

Given Ratio of sides Angle, side

Looking for Angle measure Missing side

UseSIN-1

COS-1

TAN-1

SIN, COS, TAN

Set your calculator to ‘Degree’…..

MODE (next to 2nd button)

Degree (third line down… highlight it)

2nd

Quit

Practice some more…

Find tan A: 24.19 12

A 21

Tan A = opp/adj = 12/21

Tan A = .5714

8

4A

Tan A = 8/4 = 2 8

Find tan A:

Using trig ratios in equations

Remember back in 1st grade when you had to solve:

12 = x What did you do? 6

(6) (6)

72 = xRemember back in 3rd grade when x was in

the denominator? 12 = 6 What did you do? x

(x) (x)

12x = 6__ __12 12 x = 1/2

x cm

15 cm

34°

Ask yourself:In relation to the angle,

what pieces do I have?

Opposite and hypotenuse

Ask yourself:

What trig ratio uses Opposite and Hypotenuse?

SINE

Set up the equation and solve:

Sin 34 = x 15

(15) (15)

(15)Sin 34 = x8.39 cm = x

x cm

12 cm

53°

Ask yourself:In relation to the angle,

what pieces do I have?

Opposite and adjacent

Ask yourself:

What trig ratio uses Opposite and adjacent?

tangent

Set up the equation and solve:

Tan 53 = x 12

(12) (12)

(12)tan 53 = x15.92 cm = x

x cm

18 cm

68°

Ask yourself:In relation to the angle,

what pieces do I have?

Adjacent and hypotenuse

Ask yourself:

What trig ratio uses adjacent and hypotnuse?

cosine

Set up the equation and solve:Cos 68 = 18 x

(x) (x)

(x)Cos 68 = 18

X = 18 cos 68

_____ _____cos 68 cos 68

X = 48.05 cm

Ok… we’ve found side lengths, now let’s find angle measures.

Refer to your table… what function will we use to find angle measures?

SIN-1

COS-1

TAN-1These are called INVERSE FUNCTIONS

42 cm

22 cm

θ

This time, you’re looking for theta. Ask yourself:In relation to the angle, what pieces do I have? Opposite and hypotenuse

Ask yourself:

What trig ratio uses opposite and hypotenuse? sine

Set up the equation (remember you’re looking for theta):

Sin θ = 22 42

Remember to use the inverse function when you find theta

THIS IS IMPORTANT!!

Sin -1 22 = θ 42

31.59°= θ

Let’s practice…

C

2cm

B 3cm A

Find an angle that has a tangent (ratio) of 2

3

Round your answer to the nearest degree.

Process:

I want to find an ANGLE

I was given the sides (ratio)

Tangent is opp

adj

TAN-1(2/3) = 34°

17 cm

22 cm

θ

You’re still looking for theta.

Ask yourself:

What trig ratio uses the parts I was given? tangent

Set it up, solve it, tell me what you get.

tan θ = 17 22

THIS IS IMPORTANT!!

tan -1 17 = θ 22

37.69°= θ

Your assignment

TrigWorksheets

(Kuta)