5.4 Trig. Ratios

Trigonometer

Trigon- Greek for triangles

Metric- Greek for Measure

Trig Ratios

Trig. Ratios are used to find……• A missing side on a right triangle using one

given side and an angle.

Ex:

3cm30o

Trig Ratios Continued …

Trig Ratios are also used to find…• A missing angle on a right triangle using

two given two sides.

Ex.

3cm

4cm

Identifying sides of a triangle

• Sides of a triangle are referenced to the angle

AdjacentHypotenuse

Opposite

What if the angle is in the other corner?

• Sides are identified according to the reference angle.

Opposite Hypotenuse

Adjacent

Trig Ratios Continued…

• Trig Ratios are constant values of right triangles that are based on the ratios of side measurements.

SOH CAH TOA

OPPOSITESIN

HYPOTENUSEADJACENT

COSHYPOTENUSEOPPOSITE

TANADJACENT

Trig Ratios

5cm13cm

12cm

• If you want to find the angle in degrees you can use your calculator or a chart

5sin( ) .3846

13

12cos( ) .9231

13

5tan( ) .4167

12

opp

hyp

adj

hyp

opp

adj

Finding the Angle in Degrees

• We already found this information

• Make sure your calculator is in degrees.

• Punch in 2nd Function sin(.3846)

• Your answer should be 22.62 degrees

• Punch in 2nd Function cos(.9231) and your answer should still be 22.62 degrees

• And the same thing if you do it for Tan

5sin( ) .3846

13

12cos( ) .9231

13

5tan( ) .4167

12

opp

hyp

adj

hyp

opp

adj

Finding the angle given 2 sides of a triangle

8cm12cm

What are we going to use Sin, Cos or Tan?

• We have to use Sin

WHY?• We have to use sin

because we have the opposite side and the hypotenuse

1 1

8sin( ) .6667

12

sin (sin( ) sin .6667

41.81o

Finding the angles of a triangle

• 6 15

• What ratio are we going to use sin cos or tan?

• We have to use cos because we have the adjacent side and the hypotenuse side.

sin( )

cos( )

tan( )

opp

hyp

adj

hyp

opp

adj

1 1

6cos( ) .4

15

cos cos( ) cos (.4)

23.58o

adj

hyp

Finding two angles given 2 sides

15

30

What do we use sin, cos or tan when looking for the angles?

1 1

30tan 2

15

tan tan( ) tan (2)

63.43

opp

adj

1 1

15tan( ) .5

30

tan tan( ) tan (.5)

26.56

Find the missing Angle

9 15 8

20

sin( )

cos( )

tan( )

opp

hyp

adj

hyp

opp

adj

Solutions

Example #1

Example #2

1 1

9sin( ) .6

15

sin sin( ) sin (.6)

36.87o

1 1

8tan( ) .4

20

tan tan( ) tan (.4)

21.8o

Trig Ratios Part 2

• Using Ratios to find the missing side.

y X

15 o45

You have to decide what formula you are going to use Sin, Cos or Tan

Lets use

Now solve for y by multiplying both sides by 15

hyp

oppTan

15)45(

yTan o

)15(15

)15)(45(y

Tan

x

xTan

15

15*)45(

Trig Ratios Part 2

• Using Ratios to find the missing side.

y X

15 o45

You have to decide what formula you are going to use to solve for x Sin, Cos or Tan

Lets use cos

We have to move the x to the other side if we want to solve for X so to do this we must multiply both sides by x.

The x’s will cancel on the right and now we must divide both sides by cos(45)

x

15)45cos(

XX

X *15

*)45cos(

21.21

)45cos(

15

)45cos(

15

)45cos(

*)45cos(

x

x

x

Solving for Sides VS Angles

• When solving for a side you have to use the function cos, sin, and tan

• When solving for an angle you need to use the inverse function

(On your calculator you must push 2nd function first)

111 tan,sin,cos

Solving for a Side

• Find the missing side

12

x 30

y

• What are we going to use to find x

SOH CAH TOA

x

x

x

x

6

12*)30sin(

12*12

12*)30sin(

12)30sin(

Solving for a Side

• Find the missing side

12

x 30

y

• What are we going to use to find y SOH CAH TOA

y

y

y

y

4.10

12*)30cos(

12*12

12*)30cos(

12)30cos(