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Table of Contents
5. Right Triangle Trigonometry
Right Triangle Trigonometry
Essential Question – How can
right triangles help solve real
world applications?
The Pythagorean theorem
• In a rt Δ the square of the length of the
hypotenuse is equal to the sum of the
squares of the lengths of the legs.
c2 = a2+b2
c
a
__
__
b
Example
x
7
__
__
24
x2=72+242
x2=49+576
x2=625
x=25
x 2  12 2  (6 5 ) 2
Example
x 2  144  36 * 5
x 2  144  180
x 2  36
x6
x
12
6 5
3 basic trig ratios
• Sine (sin)
• Cosine (cos)
• Tangent (tan)
Opp. side
• Sin 
hypot
adj. side
• Cos 
hypot
opp
• Tan 
adj
SOH CAH TOA
• Sin = opp/hyp
• Cos = adj/hyp
• Tan = opp/adj
Ex: Find sin, cos,
& tan of A & B.
A
sin A= 12/13
cos A= 5/13
tan A= 12/5
12
B
B
sin B= 5/13
cos B= 12/13
tan B= 5/12
13
C
5
A
Inverse Trig Functions
• Cosecant is the inverse of sin
• Secant is the inverse of cos
• Cotangent is the inverse of tan
1
hyp
csc  

sin  opp
1
hyp
sec  

cos  adj
1
adj
cot  

tan  opp
Given that sin   4 5 , calculate the other trigonometric
functions for  .
Step 1: Draw a right
triangle and find
third side using
Pythagorean theorem.
5
4

3
Step 2: Find the other ratios using formulas.
4
sin  =
5
5
csc  =
4
3
cos  =
5
5
sec  =
3
4
tan  =
3
3
cot  =
4
More examples
Given that sin  = 7/25, sketch the triangle
and find the third side. Then find cos 
Given that tan  = ¾, sketch the triangle
and find the third side. Then find sin 
Given that tan  = 4/5, what is cot  ?
Given a point, find all trig
functions
1. Draw right triangle
2. Label theta
3. Label sides
4. Use Pythagorean theorem to find missing
side
5. Find all 6 functions
Example
• Given the point (-4,10) find the values of
the six trig function of the angle.
1. Plot point
(-4,10)
2. Draw rt triangle
3. Label angle
and sides
10.8
10

4. Use Pyt. Th.
to find 3rd side.
5. Find trig
functions
-4
10
10.8
4
cos   
10.8
10
5
tan     
4
2
sin  
10.8
10
10.8
sec   
4
4
2
cot     
10
5
csc  
Example
• Given the point (-5,-2) find the values of
the six trig function of the angle.
1. Plot point
2. Draw rt triangle
3. Label angle
and sides
-5
4. Use Pyt. Th.
to find 3rd side.
-2
(-5,-2)
5. Find trig
functions
sin   
2
2 29

29
29
cos   
5
5 29

29
29
tan  
2
5

29
29
2
29
sec   
5
5
cot  
2
csc   
Last type of problem
You are given a trig ratio
It can be in one of two quadrants
Therefore you have to be given another piece
of information to determine which quadrant it
is in
Always Study Trig Carefully
Sin
Cos
Tan
y values
x values
sin/cos
Where are these positive?
Always
Sin +
Cos Tan -
Sin +
Cos +
Tan +
Sin
All
Study
Trig
Sin Cos Tan +
Sin Cos +
Tan -
Carefully
Tan
Cos
Steps
• 1. Find what quadrant the triangle is in
• 2. Draw right triangle, only sides will be negative,
hypotenuse will never be negative
• 3. Use Pythagorean theorem to find 3rd side
• 4. Find other trig functions
Example
• Given that cos θ = 8/17 and tan θ < 0,
find all six trig functions.
8
θ
-15
17
Triangle is in 4th quadrant because that is
where cos is positive and tan is negative
sin   
csc   
15
17
cos  
8
17
17
17
sec  
15
8
tan  
cot  
15
8
8
15