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
opp
SineA 
hyp
•
Adj
Cosin eA 
Hyp
Opp
TangentA 
Adj
When you Input ratios of a right triangle into the inverse
trig functions the output is the angle.
 Adj 
Opp 
Opp 
1
1
A  Sin   A  Cos 
 A  Tan 

Hyp 
Hyp 
 Adj 
1
Solving Right Triangles
A
SOLVE TRIANGLE ABC
x
Means to find the measure
of all the missing angles
and the length of all the
missing sides
5
B
2
To Find Angle A:
C
To Find Angle C: Use Tangent to find Side AB:
2
2 
1 2  SideAB : tan 24 
 A  sin   C  cos  
x
5 
5 
2
 x
tan 24
 24
 66
1
x  4.6
II. Non Right Triangles
 Law of Sine
 Law of Cosine
 Applications
Introduction to Law of Cosine
 Formula:
a 2  b 2  c 2  2bcCosA
•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
C
Formula for Law of Cosine
10
a  b  c  2bcCosA
2
2
2
a
FIND SIDE BC
(<A = 44 Degrees)
A
Stick the numbers in…
8
a  8  10  2(8)(10)Cos 44
2
2
2
B
 164  160Cos44

48.905631
a  7
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
Given
(<A = 40 Degrees)
(<C = 80 Degrees)
Formula for Law of Sine :
SinA
SinB
SinC


a
b
c
Find side a
10
A
B
Put the numbers in…
C
10( Sin 40)
Sin80 Sin 40

 a
10
a
Sin80

a  6.5
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.
Now since you know 2 sides and one angle,
you can use Cosine Law:
c  4  6  2(4)(6)Cos36
2
2
2
c 2  13.16718427
c  3.628661
B

c?
4
S
36
C
6
Short Quiz
1. Determine the length of QR to the nearest millimeter.
PQR, R  90 , Q  25 , PR  7.0cm
2. Calculate Angle R to the nearest tenth of a degree.
PQR, Q  115 , PQ  4.5cm, PR  10.8cm
3.Calculate the measure of Angle Q to the nearest degree.
PQR, RP  7, RQ  8, PQ  10
Answers
1.
tan 25 
7
QR
QR  tan 25  7
7
QR 
tan 25
QR  15.0cm
2.
sin Q sin R

q
r
sin115 sin R

10.8
4.5
4.5sin115
 sin R
10.8
0.377628  sin R
R  sin 1 (0.377628)
R  22.2
3.
q 2  p 2  r 2  2 pr cosQ
7 2  8 2  10 2  2(8)(10)cosQ
49  164 160cosQ
160cosQ  115
115
cosQ 
160
1115 
Q  cos  
160 
Q  44