R I A N G L E

Trigonometric RatiosA 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.

Finding Trig Ratios

• A trigonometric 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 trigonometric ratios are called sine, cosine, and tangent

Finding Trig Ratios

Some terminology:

• Before we can use the ratios we need to get a few terms straight

• The hypotenuse (hyp) is the longest side of the triangle – it never changes

• The opposite (opp) is the side directly across from the angle you are considering

• The adjacent (adj) is the side right beside the angle you are considering

A picture always helps…

• looking at the triangle in terms of angle b

AC

B

b

adjhyp

opp

b C is always the hypotenuse

A is the adjacent (near the angle)

B is the opposite (across from the angle)

LongestNear

Across

But if we switch angles…

• looking at the triangle in terms of angle a

AC

B

a

opphyp

adja

C is always the hypotenuse

A is the opposite (across from the angle)

B is the adjacent (near the angle)

LongestAcross

Near

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…

hypotenuse

leg

leg

In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse

a

b

c

We’ll label them a, b, and c and the angles and . Trigonometric functions are defined by taking the ratios of sides of a right triangle.

B

A

First let’s look at the three basic functions.

SINECOSINE

TANGENT

They are abbreviated using their first 3 letters

oppositesin

hypotenuse

aA

c

opposite

adjacentcos

hypotenuse

bA

c

adjacent

oppositetan

adjacent

aA

b

the trig functions of the angle B using the definitions.

a

b

c

B

ASOHCAHTOA

oppositesin

hypotenuse

bB

c

adjacentcos

hypotenuse

aB

c

oppositetan

adjacent

bB

a

opposite

adjacent

SOHCAHTOA

hypotenuse

It is important to note WHICH angle you are talking about when you find the value of the trig function.

a

bc

A

Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so

3

45

sin A = SOH-CAH-TOAh

o5

3

opposite

hypotenuse

tan B =

a

o3

4

opposite

adjacent B

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.

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

3

45

B

Oh, I'm

acute!

So am I!

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

The nice thing is that your calculator has a tan, sin and cos key that can save you some work.

However, you must remember to change the settings on your calculator from Radians to Degrees. Radians is the default setting. Here are the steps:

• Press Mode• Move the curser down to the 3rd line “Radians”• Slide the curser over so “degree” is blinking• Press 2nd Quit

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

Calculating a side if you know the angleyou know an angle (25°) and its adjacent sidewe want to know the opposite side

adj

opptan

tan 256

A

How can we use these?

A C

B = 625°

b

.47

1 6

A

6 (.47) A

2.80 A

Another example• If you know an angle and its opposite side, you can find the adjacent side.

adj

opptan

.47 6B

A = 6 C

B 25°

b

6tan 25

B

.47 6

1 B .47 6

12.76.47 .47

B

12.76B

How can we use it?Suppose we want to find an angle

and we only know two side lengthsSuppose we want to find angle a• Is side A opposite or adjacent?

• what is side B? • with opposite and adjacent we

use the…adj

opptan

A = 3C

B = 4a

b

the opposite

the adjacent

tan ratio

Lets solve it

adj

opptan

75.04

3tan a

36.87º a

A = 3C

B = 4a

b

0.75

tana

When the tan, sin or cos is in the denominator, we are going to use the reciprocal buttons. Look above tan on the calculatorYou should see TAN-1

Press 2nd TAN (.75)

Another tangent example…

• we want to find angle b• B is the opposite• A is the adjacent• so we use tan

adj

opptan

A = 3C

B = 4

a

b

13.53

33.1tan3

4tan

b

b

b

Ex. 6: Indirect Measurement• You are measuring the height of

a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.

The math

tan 59° =opposite

adjacent

tan 59° =h

45

45 tan 59° = h

45 (1.6643) ≈ h

75.9 ≈ h

Write the ratio

Substitute values

Multiply each side by 45

Use a calculator or table to find tan 59°

Simplify

The tree is about 76 feet tall.

Ex. 7: Estimating Distance

• Escalators. The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg of 76 feet.

d76 ft

30°

Now the math d76 ft

30°sin 30° =

opposite

hypotenuse

sin 30° =76

d

d sin 30° = 76

sin 30°

76d =

0.5

76d =

d = 152

Write the ratio for sine of 30°

Substitute values.

Multiply each side by d.

Divide each side by sin 30°

Substitute 0.5 for sin 30°

Simplify

A person travels 152 feet on the escalator stairs.

Why do we need the sin & cos?

• We use sin and cos when we need to work with the hypotenuse

• if you noticed, the tan formula does not have the hypotenuse in it.

• so we need different formulas to do this work• sin and cos are the ones!

C = 10A

25°

b

B

Lets do sin first

• we want to find angle a• since we have opp and hyp we use sin

hyp

oppsin

C = 10

a

b

B

A = 5

30

5.0sin10

5sin

a

a

a

And one more sin example

• find the length of side A• We have the angle and

the hyp, and we need the opp

hyp

oppsin

C = 20

25°

b

B

A 45.8

2042.0

2025sin20

25sin

A

A

A

A

And finally cos

• We use cos when we need to work with the hyp and adj

• so lets find angle bhyp

adjcos

C = 10

a

b

B

A = 4

42.66

4.0cos10

4cos

b

b

b

23.58 a

66.42 - 90 a

Here is an example• Spike wants to ride down a steel

beam• The beam is 5m long and is leaning

against a tree at an angle of 65° to the ground

• His friends want to find out how high up in the air he is when he starts so they can put add it to the doctors report at the hospital

• How high up is he?

How do we know which formula to use???

• Well, what are we working with?• We have an angle• We have hyp• We need opp• With these things we will use

the sin formula

C = 5

65°

B

So lets calculate

• so Spike will have fallen 4.53m

C = 5

65°

B

53.4

591.0

565sin5

65sin

65sin

opp

opp

opp

opp

hyp

opp

One last example…

• Lucretia drops her walkman off the Leaning Tower of Pisa when she visits Italy

• It falls to the ground 2 meters from the base of the tower

• If the tower is at an angle of 88° to the ground, how far did it fall?

First draw a triangle

• What parts do we have?• We have an angle• We have the Adjacent• We need the opposite• Since we are working with

the adj and opp, we will use the tan formula

2m

88°

B

So lets calculate

• Lucretia’s walkman fell 57.27m

2m

88°

B

27.57

264.28

288tan2

88tan

88tan

opp

opp

opp

opp

adj

opp

An application

65°

10m

You look up at an angle of 65° at the top of a tree that is 10m away

the distance to the tree is the adjacent side the height of the tree is the opposite side

4.21

14.210

65tan1010

65tan

opp

opp

opp

opp

What are the steps for doing one of these questions?

1. Make a diagram if needed2. Determine which angle you are working with3. Label the sides you are working with4. Decide which formula fits the sides5. Substitute the values into the formula6. Solve the equation for the unknown value7. Does the answer make sense?

Two Triangle Problems

• Although there are two triangles, you only need to solve one at a time

• The big thing is to analyze the system to understand what you are being given

• Consider the following problem:• You are standing on the roof of one building

looking at another building, and need to find the height of both buildings.

Draw a diagram

• You can measure the angle 40° down to the base of other building and up 60° to the top as well. You know the distance between the two buildings is 45m

60°

40°

45m

Break the problem into two triangles.

• The first triangle:

• The second triangle

• note that they share a side 45m long

• a and b are heights!

60°

45m

40°

b

a

The First Triangle

• We are dealing with an angle, the opposite and the adjacent

• this gives us Tan

60°

45m

a

77.94m a

451.73a

4560tan45

60tan

a

a

The second triangle

• We are dealing with an angle, the opposite and the adjacent

• this gives us Tan

45m

40°

b

37.76mb

450.84b

4540tan45

40tan

b

b

What does it mean?

• Look at the diagram now:• the short building is

37.76m tall• the tall building is 77.94m

plus 37.76m tall, which equals 115.70m tall

60°

40°

45m

77.94m

37.76m

Ex: 5 Using a Calculator

• You can use a calculator to approximate the sine, cosine, and the tangent of 74. Make sure that your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.