Warm Up 1. What is the third angle measure in a triangle with angles measuring 65° and 43°? Find each value. Round trigonometric ratios to the nearest hundredth and angle measures to the

nearest degree.

2. sin 73° 3. cos 18° 4. tan 82°

5. sin-1 (0.34) 6. cos-1 (0.63) 7. tan-1 (2.75)

72°

0.96 0.95 7.12

20° 51° 70°

Example 1: Finding Trigonometric Ratios for Angles

Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.

A. tan 103° B. cos 165° C. sin 93°

tan 103° ! –4.33

cos 165° ! –0.97

sin 93° ! 1.00

Example 2: Finding Trigonometric Ratios for Angles

Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.

A. tan 103° B. cos 165° C. sin 93°

tan 103° ! –4.33

cos 165° ! –0.97

sin 93° ! 1.00

More Examples

Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.

a. tan 175°

tan 175° ! –0.09

b. cos 92° c. sin 160°

cos 92° ! –0.03 sin 160° ! 0.34

Applying the Primary Trigonometric Ratios

In trigonometry, the ratio we are talking about is the comparison of the sides of a RIGHT TRIANGLE.

Two things MUST BE understood: 1. This is the hypotenuse.. This

will ALWAYS be the hypotenuse 2. This is 90°! this makes the

right triangle a right triangle!. Without it, we can not do this trig! we WILL NOT use it in our calculations because we COULD NOT do calculations without it.

Now that we agree about the hypotenuse and right angle, there are only 4 things left; the 2 other

angles and the 2 other sides.

A We will refer to the sides in terms of their proximity to the angle

If we look at angle A, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse.

opposite

adjacent hypotenuse

B

If we look at angle B, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse.

opposite

adjacent

hypotenuse

Remember we won’t use the right angle

X

" 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!

To Remember our Trigonometric Ratios we can think of the

following:

SohCahToa

! There are 3 kinds of trigonometric ratios we will learn.

! sine ratio

! cosine ratio

! tangent ratio

Three Types Trigonometric Ratios

Trigonometric Ratios Name “say”

Sine Cosine tangent

Abbreviation Abbrev.

Sin Cos Tan

Ratio of an angle measure

Sin" = opposite side hypotenuse

cos" = adjacent side hypotenuse

tan" =opposite side adjacent side

Primary Trigonometric Ratios !  Sine, Cosine, and Tangent are the primary

trigonometric ratios that are used to solve for finding the unknown side of a right angle triangle

!  Primary Trigonometric Ratios

b C

B

A

c a

SinA = ac

CosA = bc

baTanA =

Side opposite of angle A

Hypotenuse

Side adjacent of angle A

hypopp

hypadj

adjopp

SOH CAH TOA

Definitions

!  Angle of Elevation – The angle between the horizontal and the line of sight when one is looking up at an object

!  Angle of Depression – the angle between the horizontal and the line of sight when one is looking down at an object

Example

!  Determine the length of m in Triangle MNP.

N

M

P

225 ft

60

•  The length of the hypotenuse is given •  The measure of the acute angle P is given •  m is adjacent to angle P •  What Trigonometric ratio will we use? •  Cosine

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m

Solution ! Using the Cosine ratio

! Therefore, the length of m is about 112.5 feet

N

M

P

225 ft

60

CosP = NPPM

= mn

22560 m

Cos =

mCos =! 22560

m =112.5m

Example

! Determine the height of the Eiffel Tower if one is standing 68 m from the base and the angle of elevation to the top is 78 degrees.

Solution

Base

Top of Tower

Person 78 68

Height

TanP = OppAdj

Tan78 = Opp68

68(Tan78) = Opp

319.9 = OppTherefore, the height of the Eiffel Tower is 319.9 m

Inverse Trigonometric Ratios !  The inverse of Sine, Cosine, and Tangent are used to

solve for the unknown angle of elevation or depression of a right angle triangle

!  Inverse Trigonometric Ratios

b C

B

A

c a

caSinA 1!="

cb

CosA 1!="

baTanA 1!="

Side opposite of angle A

Hypotenuse

Side adjacent of angle A

Example

! Find angle P using the proper inverse ratio

•  The length of the hypotenuse is given

•  The opposite measurement of angle P is given

•  What Trigonometric ratio will we use?

•  Sine

N

M

P

5 m 3m

Solution

Therefore, Angle P has an angle of approximately of 37 degrees.

SinP = pn

SinP = 35

SinP = 0.6

!P = Sin"1(0.6)

!P = 36.87 " 37

N

M

P

5 m 3m

!P = 0.6Sin

Remember 1/a equals

a-1

Example

! The Empire State Building height is 381.32 m. What is the angle of elevation if a person is 267 m away from the base of the building?

Solution

Base

Top of Building

Person P 267

381.32

TanP = OppAdj

TanP = 381.32267

!P = 1.428Tan

Therefore, angle of elevation is approximately 55 degrees

TanP =1.428

!P = Tan"1(1.428)

!P = 54.99 " 55

Make sure you have a calculator! Given Ratio of sides Angle, side

Looking for Angle measure Missing side

Use SIN-1

COS-1 TAN-1

SIN, COS, TAN

Set your calculator to ‘Degree’!..

MODE (next to 2nd button)

Degree (third line down! highlight it)

2nd

Quit