Chapter 5 Introduction to Trigonometry: 5.7 The Primary

Trigonometric Ratios

Humour Break

5.7 The Primary Trigonometric Ratios

Goals for Today:• Learn how to identify sides and angles in

triangles• Learn the primary trigonometric ratios• Use the primary trigonometric ratios to find a

missing side• Use the primary trigonometric ratios to find a

missing angle

5.7 Primary Trigonometric RatiosIn previous math course, you learned how to

calculate a missing side in a right angle triangle when you were given two other sides.

You did this using the pythagorean theorem

a² + b² = c²

5.7 Primary Trigonometric RatiosYou also learned to how to calculate missing

angles using angle rules, such as the 180° triangle rule, where the sum of angles in a triangle add up to 180 °

5.7 Primary Trigonometric RatiosNow, we are going to introduce the

trigonometric ratios, where we work with both angles and sides to find unknown angles in sides.

Today, we are working with the primary trig ratios which are used for right angle triangles only

5.7 Primary Trigonometric Ratios

What are the sides from the perspective of angle A?

5.7 Primary Trigonometric Ratios

Opposite (or side “a”)

From the perspective of angle A....

Adjacent (or side “b”)

Hypotenuse (or side “c”)

5.7 Primary Trigonometric Ratios

What are the sides from the perspective of angle B?

5.7 Primary Trigonometric Ratios

Adjacent* (or side “a”)

From the perspective of angle B....

Opposite* (or side “b”)

Hypotenuse (or side “c”)

* Side name changes because from the perspective of a different triangle

5.7 Primary Trigonometric RatiosFor a right angle triangle, the three primary trig

ratios are:

adjacentoppositeA

hypotenuseadjacentA

hypotenuseoppositeA

tan

cos

sin

5.7 Primary Trigonometric RatiosAn acronym to help remember these formulas is

SOHCAHTOA

5.7 Primary Trigonometric Ratios

With these ratios you are dealing with one of two situations

(1) You have two sides in a triangle and you use them to find an angle by using the inverse or 2nd of SIN, COS, or TAN on your calculator

(2) You have a side and an angle in a triangle and you want to find another side by using the SIN, COS, or TAN on your calculator

5.7 Primary Trigonometric RatiosConsider the classic 3, 4, 5 right triangle

5.7 Primary Trigonometric RatiosLet’s use scenario 1 to find angle A. Which ratio?

5.7 Primary Trigonometric RatiosSIN because side a = 3 (opposite) and side c = 5

(hypotenuse) from angle A

5.7 Primary Trigonometric RatiosSIN A = ∠ opposite hypotenuse

reesAASin

ASin

HypotenuseOppositeASin

deg9.366.053

Hint: Once you have the trig ratio, you use your SIN to -1 or inverse function to convert the ratio 0.6 to the angle of 36.9°

5.7 Primary Trigonometric RatiosYour calculator does a nice job. In the old days

(when I was in high school) you had to look up the ratio in a table and convert it into an angle!

5.7 Primary Trigonometric RatiosLet’s again use scenario 1 to find angle A. Which

ratio? Why did it change?

5.7 Primary Trigonometric RatiosCOS because side b = 4 (adjacent) and side c = 5

(hypotenuse) from angle A

5.7 Primary Trigonometric RatiosCOS A = ∠ adjacent hypotenuse

reesAACos

ACos

HypotenuseAdjacentACos

deg9.368.054

Hint: Once you have the trig ratio, you use your COS to -1 or inverse function to convert the ratio 0.8 to the angle of 36.9°We got the same angle which makes sense!... Same triangle!

5.7 Primary Trigonometric RatiosFinally, let’s again use scenario 1 to find angle A.

Which ratio?

5.7 Primary Trigonometric RatiosTAN because side b = 4 (adjacent) and side a = 3

(opposite) from angle A

5.7 Primary Trigonometric RatiosTAN A = ∠ opposite adjacent

reesAACos

ATan

AdjacentOppositeATan

deg9.3675.043

Hint: Once you have the trig ratio, you use your TAN to -1 or inverse function to convert the ratio 0.75 to the angle of 36.9°We got the same angle which makes sense!... Same triangle!

5.7 Primary Trigonometric RatiosLet’s use scenario 1 to find angle B. Which ratio? What’s different than when we were finding angle A?

5.7 Primary Trigonometric Ratios

When we were finding angle A, given these sides, we used the SIN ratio. From angle B we are dealing with the adjacent sides and the hypotenuse so we have to use the COS ratio.

5.7 Primary Trigonometric RatiosCOS B = ∠ adjacent hypotenuse

reesBBCos

BCos

HypotenuseAdjacentBCos

deg1.536.053

Hint: Once you have the trig ratio, you use your COS to -1 or inverse function to convert the ratio 0.6 to the angle of 53.1°

5.7 Primary Trigonometric RatiosLet’s use scenario 1 to find angle B. Which ratio? What’s different than when we were finding angle A?

5.7 Primary Trigonometric Ratios

When we were finding angle A, given these sides, we used the COS ratio. From angle B we are dealing with the opposite sides and the hypotenuse so we have to use the SIN ratio.

5.7 Primary Trigonometric RatiosSIN B = ∠ opposite hypotenuse

reesBBSin

BSin

HypotenuseOppositeBSin

deg1.538.054

Hint: Once you have the trig ratio, you use your SIN to -1 or inverse function to convert the ratio 0.6 to the angle of 36.9°

5.7 Primary Trigonometric RatiosLet’s use scenario 1 to find angle B. Which ratio? What’s different than when we were finding angle A?

5.7 Primary Trigonometric Ratios

When we were finding angle A, given these sides, we used the TAN ratio. From angle B we are dealing with the opposite sides and the adjacent sides again, so we still use the TAN ratio, but the numbers are reversed

5.7 Primary Trigonometric RatiosTAN A = ∠ opposite adjacent

reesBBCos

BTan

AdjacentOppositeBTan

deg1.533333.134

Hint: Once you have the trig ratio, you use your TAN to -1 or inverse function to convert the ratio 1.3333 to the angle of 53.1°We got the same angle which makes sense!... Same triangle!

5.7 Primary Trigonometric RatiosNow, lets again consider the classic 3, 4, 5 right triangle,

but this time, given an angle and a side and asked to find a side

5.7 Primary Trigonometric RatiosFrom the perspective of angle A, we are dealing with

the opposite side and the hypotenuse… so we have to use the SIN ratio…

5.7 Primary Trigonometric RatiosNow, lets again consider the classic 3, 4, 5 right triangle,

but this time, given angle A and side c and asked to find a side a…

This is scenario 2… given an angle and a side… find another side…

5.7 Primary Trigonometric RatiosSIN A = ∠ opposite hypotenuse

3002.3

55

6004.05

56004.0

5deg9.36

aa

xax

a

areesSin You input 36.9 into your calculator and hit the SIN button to get the ratio 0.6004. In some calculators, the order is reversed…

5.7 Primary Trigonometric RatiosSimiliarly, if asked to side side b, from the perspective of angle A,

we are here dealing with the adjacent side side and the hypotenuse… which ratio would we use?

5.7 Primary Trigonometric RatiosThe COS ratio because from the perspective of angle A, we are

dealing with the adjacent side and the hypotenuse…

5.7 Primary Trigonometric RatiosCOS A = ∠ adjacent hypotenuse

49985.3

55

7997.05

57997.0

5deg9.36

bb

xbx

b

breesCos You input 36.9 into your calculator and hit the COS button to get the ratio 0.7997. In some calculators, the order is reversed…

5.7 Primary Trigonometric RatiosSimiliarly, if asked to find side b, from the perspective of angle A,

but we were given side a… we sould be dealing with the opposite side and the adjacent side side… which ratio would we use?

5.7 Primary Trigonometric RatiosThe Tan ratio, because we are dealing with the opposite and

adjacent sides…

5.7 Primary Trigonometric RatiosTan A = ∠ opposite adjacent

49957.3

7508.03

7508.07508.0

37508.0

))(3()7508.0)((

37508.0

3deg9.36

bb

b

b

bb

b

b

breesTan You input 36.9 into

your calculator and hit the TAN button to get the ratio 0.7508. In some calculators, the order is reversed…

5.7 Primary Trigonometric RatiosIf we now examine things from the perspective of angle B, we

are dealing with the adjacent side and the hypotenuse… so we have to use the COS ratio…

We are again dealing with Scenario 2… given a side and an angle, finding another side…

5.7 Primary Trigonometric RatiosCOS B = ∠ adjacent hypotenuse

3002.3

55

6004.05

56004.0

5deg1.53

aa

xax

a

areesCos You input 53.1 into your calculator and hit the COS button to get the ratio 0.6004. In some calculators, the order is reversed… Note that COS 53.1° is the same as SIN 36.9°

5.7 Primary Trigonometric RatiosIn this next example… again from the perspective of angle B, we

are dealing with the opposite side and the hypotenuse… so we have to use the SIN ratio…

We are again dealing with Scenario 2… given a side and an angle, finding another side…

5.7 Primary Trigonometric RatiosSIN B = ∠ opposite hypotenuse

49985.3

55

7997.05

57997.0

5deg1.53

bb

xbx

b

breesSin You input 53.1 into your calculator and hit the SIN button to get the ratio 0.7997. In some calculators, the order is reversed…

5.7 Primary Trigonometric RatiosIn this final example… again from the perspective of angle B, we

are dealing with the opposite side and the adjacent side… so we have to use the TAN ratio…

We are again dealing with Scenario 2… given a side and an angle, finding another side…

5.7 Primary Trigonometric RatiosTAN B = ∠ opposite adjacent

49957.3

33

3319.13

33319.1

3deg1.53

bb

xbx

b

breesTan You input 53.1 into your calculator and hit the SIN button to get the ratio 0.7997. In some calculators, the order is reversed…

Homework

• Tuesday, December 3rd – p.496, #1, 2, 4, 9-11• Thursday, December 12th – p.498, #12-15, 17,

19-21• Friday, December 13th – p.498, #22-26