Introduction to Trigonometry

This section presents the 3 basic trigonometric ratios sine, cosine, and tangent. The concept of similar triangles and the Pythagorean Theorem can be used to develop the trigonometry of right triangles.

Engineers and scientists have found it convenient to formalize the relationships by naming the ratios of the

sides.

You will memorize these 3 basic ratios.

The Trigonometric Functions

SINE

COSINE

TANGENT

SINEPronounced like “sign”

Pronounced like “co-sign”

COSINE

Pronounced “tan-gent”

TANGENT

HypA toOpp

Sin(A)

HypA toAdj

Cos(A)

A toAdjA toOpp

Tan(A) A

B

C

With Respect to angle A, label the three sides

We need a way to remember all of these ratios…

SOHCAHTOA

SinOppHypCosAdjHypTanOppAdj

Finding sin, cos, and tan.

(Just writing a ratio or decimal.)

Find the sine, the cosine, and the tangent of M.

Give a fraction and decimal answer (round to 4 places).

8.10

9 8333.

8.10

6 5556.

6

9 5.1

9

6

10.8

MP

N

hypopp

Msin

hypadj

M cos

adjopp

Mtan

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

24.5

23.1

8.2

hyp

oppA sin

5.24

2.8 3347.

hyp

adjA cos

5.24

1.23 9429.

adj

oppA tan

1.23

2.8 3550.

Give a fraction and decimal answer.

Round to 4 decimal places

B

C

Finding a side.(Figuring out which ratio to use

and getting to use a trig button.)

Ex: 1 Find x. Round to the nearest tenth.

adjopp

tan

55

20 m

x

20

55tanx

m 6.28x

x55tan20 tan 20 55 )

Figure out which ratio to use.

What we’re looking for…

opp

What we know… adj

We can find the tangent of 55 using a calculator

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

24

283 mx 283

24sinx

m 1.115x

x24sin283

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

20 m40

x

2040cos

x

m 3.15x

x40cos20

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

72

80 m

x

x8072tan

26.0 mx

8072tan x

)72tan(80x

tan 80 72 = )

Note: When the variable isin the denominator,you end up dividing

hidingSometimes the right triangle is

ABC is an isosceles triangle as marked. Find sin C.

A

13 13

10

sinopp

Chyp

12

13

Answer as a fraction.

B C

5

12

A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge?

200

x

Ex. 5

60°

cos 60°

x (cos 60°) = 200

x

X = 400 yards

A surveyor is standing 50 metres from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree?

50 m

?

tan 71.5°

tan 71.5°50

y

50 (tan 71.5°) = y

Ex: 6

Opp

Adj

y 149.4 m

71.5°

For some applications of trig, we need to know these

meanings:angle of elevation and

angle of depression.

Angle of ElevationIf an observer looks UPWARD

toward an object, the angle the line of sight makes with the horizontal.

Angle of elevation

Angle of Elevation

Angle of DepressionIf an observer looks DOWNWARD

toward an object, the angle the line of sight makes with the horizontal.

Angle of depression

Finding an angle.(Figuring out which ratio to use and

getting to use the 2nd button and one of the trig buttons. These are the

inverse functions.)

Ex. 1: Find . Round to four decimal places.

9

17.2

Make sure you are in degree mode (not radians).

1

17.2tan

917.2

tan9

62.4

2nd tan 17.2 9 )

Ex. 2: Find . Round to three decimal places.

23

7

Make sure you are in degree mode (not radians).

1

7cos

237

cos23

72.3

2nd cos 7 23 )

Ex. 3: Find . Round to three decimal places.

400

200

Make sure you are in degree mode (not radians).

1

200sin

400200

sin400

30

2nd sin 200 400 )

When we are trying to find a sidewe use sin, cos, or tan.

When we need to find an angle we use sin-1, cos-1, or tan-1.