906030 2:3:1

Short Leg:Long Leg:Hypotenuse

xxx 2:3:

30-60-90 Right Triangle

12

30

60

This is our reference triangle for the 30-60-90 triangle.

3

We will use a reference triangle to set up a proportion then solve.

600

300

x

24

Solve for x and yEx: 2

2a

a

a√3y

12 3 y

904545 2:1:1

Leg:Leg:Hypotenuse2:: xxx

x

345

EX: 3 Solve for x

x 3 2

2

a√2a

a

oppositehypotenuse

SinOpp

Hyp

adjacent

CosAdj

Hyp

TanOpp

Adj

hypotenuseopposite

adjacent

Finding a side.(Figuring out which ratio to use

and getting to use a trig button.)

Ex: 1 Figure out which ratio to use. Find x. Round to the nearest tenth.

5520 m

x

20

55tanx

m 6.28x

x55tan20tan 20 55 )

Shrink yourself down and stand where the angle is.

Now, figure out which trig ratio you have and set up the problem.

Ex: 2 Find the missing side. Round to the nearest tenth.

72

80 ft

x

x

8072tan

ft 26x

8072tan x

72tan

80x

tan 80 72 = ( ) )Shrink yourself down and stand where the angle is.Now, figure out which trig ratio you have and set up the problem.

Ex: 3 Find the missing side. Round to the nearest tenth.

24

283 mx 283

24sinx

m 1.115x

x24sin283Shrink yourself down and stand where the angle is.

Now, figure out which trig ratio you have and set up the problem.

Ex: 4 Find the missing side. Round to the nearest tenth.

4020 ft x

2040cos

x

ft 3.15x

x40cos20

Problem-Solving Strategies

You are given all 3 sides of the triangle.

Find the two non-right angles.

1. Use 2 different trig ratios to get each of the angles.

AC

B

24

257

1 1

24 24

25 724 24

25 7

16.3 73.7

CosA TanB

A Cos B Tan

A B

Angle of Elevation/Depression

Balloon

You

Angle of depression

Angle of elevation

Sometimes when we use right triangles to model real-life situations, we use the terms angle of elevation and angle of depression.

If you are standing on the ground and looking up at a hot air balloon, the angle that you look up from ground level is called the angle of elevation.

If someone is in the hot air balloon and looks down to the ground to see you, the angle that they have to lower their eyes, from looking straight ahead, is called the angle of depression.

Angle of Elevation/DepressionIf you look up 15º to see the balloon, then the person in the balloon has to look down 15º to see you on the ground.

Notice that in this situation, the one of the legs that forms the right angle is also the height of the balloon.

Angle of elevation = Angle of depression.

Balloon

You

Angle of depression = 15º

Angle of elevation= 15º

Draw a PictureWhen solving math problems, it can be very helpful to draw a picture of the situation if none is given.

Here is an example.

Find the missing sides and angles for Triangle FRY. Given that angle Y is the right angle, f = 68, and y = 88.

RY

F

68

88r

The picture helps to visualize what we know and what we want to find!