Lesson 15: Trigonometric Ratios

The Trigonometric Functions we will be

looking at

SINE

COSINE

TANGENT

The Trigonometric Functions

SINE

COSINE

TANGENT

SINE

Pronounced “sign”

Pronounced “co-sign”

COSINE

Pronounced “tan-gent”

TANGENT

Prounounced “theta”

Greek Letter

Represents an unknown angle

oppositehypotenuse

LegOppSin

Hyp

adjacent tan

Opp LegAdj Leg

hypotenuseopposite

adjacent

LegAdjCos

Hyp

SOH-CAH-TOA

S - SineO - OppositeH – Hypotenuse

C - CosineA – AdjacentH – Hypotenuse

T – TangentO - OppositeA - Adjacent

Finding sin, cos, and tan.

(Just writing a ratio.)

1. Circle Angle.

2. Label sides (hyp, opp, adj)

3. Set up ratio

Find the sine, the cosine, and the tangent of angle A.

Give a fraction.

hyp

oppA sin

8.10

9

hyp

adjA cos

8.10

6

adj

oppA tan

6

9 3

2

9

6

10.8

A

Shrink yourself down and stand where the angle is.Now, figure out your ratios.

Find the sine, the cosine, and the tangent of angle A

A

24.5

23.1

8.2

hyp

oppA sin

5.24

2.8

hyp

adjA cos

5.24

1.23

adj

oppA tan

1.23

2.8

Shrink yourself down and stand where the angle is.Now, figure out your ratios.

1. Set up and calculate the trig ratios for sine, cosine and tangent of angle A.

2. Set up and calculate the trig ratios for sine, cosine and tangent of angle B.

A carpenter needs to measure the length of the beams that will support a roof. The building is 50 feet wide. The roof will rise at an angle of 30º from the top of the walls. The peak of the roof is 14.43 feet above the top of the walls. The side adjacent to the 30º angle is half the width of the building. How long is each supporting beam, b, to the nearest thousandths? Add 2 feet to the beam length so that the roof can extend 2 feet past the walls. What is the cosine of 30º?

A water park has a straight slide into a deep pool. The slide is 50 feet long. It rises from the pool at an angle of 45º. How tall do the vertical supports of the platform need to be to support the platform at the top of the slide? The bottom of the slide is 35.355 feet from the vertical supports. What is the sine of 45º? Round to the nearest thousandths.

Students are having a contest to see who can find the tallest tree in a park. To win, a student must measure the height of the tree without climbing the tree. Martha locates a very tall oak tree. She measures that the tree’s shadow is 45 feet long. Martha has a shadow that is 11.5 feet long. She is 5.75 feet tall. How tall is the oak tree? What is the tangent of ∠A?

Lesson 15: Finding Missing Sides and Angles

Finding an angle.

(Figuring out which ratio to use and an inverse trig

button.)1. Circle Angle.

2. Label sides (hyp, opp, adj)

3. Set up ratio

4. Calculate using inverse trig

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

20 m

40 m

20tan

40

26.6o

1 20tan

40x

Tan-1 20 / 40 )

Shrink yourself down and stand where the angle is.

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

x

Ex: 2 Figure out which ratio to use. Find x. Round to the nearest tenth.

15 m50 m 15

sin50

17.5 o

1 15sin

50x

Sin-1 15 / 50 )

Shrink yourself down and stand where the angle is.

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

x

Ex. 3: Find . Round to the nearest degree.

9

17.29

2.17tan

62

1 17.2tan

9

Ex. 4: Find . Round to the nearest degree.

23

7

23

7cos

72

1 7cos

23

Ex. 5: Find . Round to the nearest degree.

400

200

400

200sin

30

1 200sin

400

Finding a side.(Figuring out which ratio to use and getting to use a trig

button.)1. Circle Angle.

2. Label sides (hyp, opp, adj)

3. Set up ratio

4. Cross Multiply

Ex: 6 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: 7 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: 8 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: 9 Find the missing side. Round to the nearest tenth.

4020 ft x

2040cos

x

ft 3.15x

x40cos20

When we are trying to find a

side we use sin, cos, or tan.

When we are trying to find an angle we use (INVERSE) sin-1, cos-1, or tan-1.