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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.
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