Download 9.5 Addition, Subtraction, and Complex Fractions

Table of Contents
3. 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
Pythagorean Theorem
• We are going to rewrite the Pythagorean
theorem for the special right triangles in a
unit circle
• x 2 + y2 = 1
• We can also rewrite this using the sin/cos
relationship on the unit circle
• cos2 + sin2 = 1
• This is called a Pythagorean trig identity
(more on this later!)
1st type of problem
Given a trig function (assuming 1st quadrant), find other
5 trig functions
Step 1: Use the Pythagorean trig identity to find sin or cos
Step 2: Find the other ratios using what we learned about
trig ratios
Given that sin   4 5, calculate the other trigonometric
functions for  .
Step 1: Use Pythagorean
trig identity to find cos
cos  =
3
5
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, find cos 
Given that tan  = ¾, find sin 
2nd type - 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
2 𝟐𝟗
10

4. Use Pyt. Th.
to find 3rd side.
5. Find trig
functions
4
10
5
5 29


29
2 29
29
4
2
2
cos   


2 29
29
29
10
5
tan     
4
2
sin  
29
5
 29
sec  
2
2
cot   
5
csc  
Example
• Given the point (-5,-2) find the values of
the six trig function of the angle. 2
2 29
sin   
1. Plot point
2. Draw rt triangle
cos   
3. Label angle
and sides
2
tan  
5
5
4. Use Pyt. Th.
to find 3rd side.
2
(-5,-2)
5. Find trig
functions

29
29

29
5
5 29

29
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. Use Pythagorean trig identity to find sin or cos
• 3. Find other trig functions remembering which are
positive and negative based on the quadrant
Example
• Given that cos θ = 8/17 and tan θ < 0,
find all six trig functions.
Triangle is in 4th quadrant because that is
where cos is positive and tan is negative
15
8
cos  
17
17
tan  
17
17
sec  
15
8
cot  
sin   
csc   
15
8
8
15
Assessment
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