5.13 Solving Triangles with Trigonometry

#(1,2,4,8,11,15)

I. Right Triangles SOH CAH TOA helps you remember which trig function to use in order to find

sides or angles of a right triangle. When you Input angles into the primary trig functions, the output is the ratio

of the sides of a right triangle

SineAopp

hyp

CosineAAdj

Hyp

TangentA Opp

Adj

ASin 1 Opp

Hyp

ACos 1 Adj

Hyp

ATan 1 Opp

Adj

• When you Input ratios of a right triangle into the inverse trig functions the output is the angle.

Solving Right Triangles

A

B C

C cos 1 2

5

66

Asin 1 2

5

24

xSideAB

224tan:

x24tan

2

6.4x

2

5

SOLVE TRIANGLE ABCSOLVE TRIANGLE ABC

Use Tangent to find Side AB:Use Tangent to find Side AB:To Find Angle A:To Find Angle A: To Find Angle C:To Find Angle C:

Means to find the measure of all the missing angles and the length of all the missing sides

x

II. Non Right Triangles

Law of Sine Law of Cosine Applications

Introduction to Law of Cosine Formula:

bcCosAcba 2222

•The small letters stand for the sides (a,b,c), While the capital letter stands for the angle (A)

•Use law of cosine when you are not given a right triangle, and you have two sides, and one angle that is in between the two sides

Law of Cosine

bcCosAcba 2222

A

B

C

a 2 82 102 2(8)(10)Cos44

905631.48

a7

44160164 Cos

Formula for Law of CosineFormula for Law of Cosine

(<A = 44 Degrees)

10

8

FIND SIDE BCFIND SIDE BC

Stick the numbers in…Stick the numbers in…

a

Introduction to Law of Sine Formula:

SinA SinB SinC

a b c

•Like law of cosine, the lower case letters represent the sides, while capital letters represent the angles

•Use law of Sine when you have an angle and a side that are opposite to each other.

Law of Sine

A B

C

SinA SinB SinC

a b c

80 40

10

Sin Sin

a

10( 40)

80

Sina

Sin

6.5a

Formula for Law of Sine :Formula for Law of Sine :

(<C = 80 Degrees)

10

(<A = 40 Degrees) Find side aFind side a

Put the numbers in…

Given

Tips & Proven Strategies

•Use law of sine when you have pairs.

•Use law of cosine when you have no pairs.

•Remember to draw diagrams to help you.

•When several steps are involved, don’t round the numbers. Keep the original, and round at the very end. By doing this, your answer will be more accurate.

Problem Solving With Trig

The Rays are under new coaching, they’ve been told that a sure to win technique is to block goals in precise triangles. If Bertuzzi and Cloutier are 4 feet away from each other, and Cloutier and Sopel are 6 feet away, how far is Sopel from Bertuzzi, if Cloutier is 36 degrees away from them both?

Remember to first draw a diagram, and label the sides with information you know.

C

S

BNow since you know 2 sides and one angle, you can use Cosine Law:

4

636

c?36)6)(4(264 222 Cosc

16718427.132 c

3.628661c

Short Quiz

cmPRQRPQR 0.7,25,90,

cmPRcmPQQPQR 8.10,5.4,115,

1. Determine the length of QR to the nearest millimeter.

2. Calculate Angle R to the nearest tenth of a degree.

10,8,7, PQRQRPPQR3.Calculate the measure of Angle Q to the nearest degree.

Answers

cmQR

QR

QR

QR

0.1525tan

7

725tan

725tan

sinQq

sinRr

sin115

10.8

sinR

4.54.5sin115

10.8sinR

0.377628 sinR

Rsin 1(0.377628)

R22.2

1.

2.

q2 p2 r2 2prcosQ

72 82 102 2(8)(10)cosQ

49 164 160cosQ

160cosQ115

cosQ115

160

Qcos 1 115

160

Q44

3.