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Table of Contents3. 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.
__ __
a
b
cc2 = a2+b2
Pythagorean Theorem
• We are going to rewrite the Pythagorean theorem for the special right triangles in a unit circle
• x2 + 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
Given a trig function (assuming 1st quadrant), find other 5 trig functions
Step 2: Find the other ratios using what we learned abouttrig ratios
Step 1: Use the Pythagorean trig identity to find sin or cos
1st type of problem
Given that , calculate the other trigonometric functions for .
Step 2: Find the other ratios using formulas.
sin =5
4
tan =3
4
sec =3
5cos =
5
3
cot =4
3
csc =4
5
4sin 5
Step 1: Use Pythagorean trig identity to find cos
cos =5
3
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 triangle2. Label theta3. Label sides4. Use Pythagorean theorem to find missing
side5. Find all 6 functions
Example• Given the point (-4,10) find the values of
the six trig function of the angle.
(-4,10)1. Plot point
2. Draw rt triangle
3. Label angleand sides
10
44. Use Pyt. Th.to find 3rd side.
2
5. Find trig functions
10 5 5 29sin
292 29 294 2 2
cos2 29 29 2910 5
tan4 2
29csc
5
29sec
22
cot5
Example• Given the point (-5,-2) find the values of
the six trig function of the angle.
(-5,-2)
1. Plot point
2. Draw rt triangle
3. Label angleand sides
2
5
4. Use Pyt. Th.to find 3rd side.
5. Find trig functions
2 2 29sin
2929
5 5 29cos
29292
tan5
29csc
2
29sec
55
cot2
29
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 pieceof information to determine which quadrant it is in
Always Study Trig Carefully
Sin +Cos +Tan +
Where are these positive?
Sin +Cos -Tan -
Sin -Cos -Tan +
Sin -Cos +Tan -
AllSin
Tan Cos
Always
Study
Trig
Carefully
Sin y valuesCos x valuesTan sin/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 iswhere cos is positive and tan is negative
17 17 8csc sec cot
15 8 15
15 8 15sin cos tan
17 17 8
Assessment
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