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 ID: 778777
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Slide1
Table of Contents3. Right Triangle Trigonometry
Slide2Right Triangle Trigonometry
Essential Question – How can right triangles help solve real world applications?
Slide3The 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
c
c
2
= a
2
+b
2
Slide4Pythagorean TheoremWe are going to rewrite the Pythagorean theorem for the special right triangles in a unit circle
x
2
+ y
2
= 1
We can also rewrite this using the sin/cos relationship on the unit circlecos2 + sin
2 = 1This is called a Pythagorean trig identity (more on this later!)
Slide5Given
a trig function (assuming 1
st
quadrant), find other
5
trig functions
Step 2: Find the other ratios using
what we learned about
t
rig ratios
Example: Six Trig Ratios
Step 1: Use the Pythagorean trig identity to find sin or cos1st type of problem
Slide6Given that , calculate the other trigonometric functions for
.
Step 2: Find the other ratios using formulas.
sin
=
tan
=
sec
=
cos
=
cot
=
csc
=
Example: Six Trig Ratios
Step 1:
Use Pythagorean trig identity to find cos
cos
=
Slide7More examples
Given that sin = 7/25,
find
cos
Given that tan = ¾,
find
sin
Slide82nd 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
Slide9Example
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 angle
and sides
10
4
4. Use Pyt. Th.
to find 3
rd
side.
2
5. Find trig
functions
Slide10Example
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 angle
and sides
2
5
4. Use Pyt. Th.
to find 3
rd
side.
5. Find trig
functions
Slide11Last 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
Slide12Always Study Trig Carefully
Sin +
Cos +
Tan +
Where are these positive?
Sin +
Cos -
Tan -
Sin -
Cos -
Tan +Sin -Cos +Tan -
AllSin
TanCos
A
lways
S
tudy
T
rig
C
arefully
Sin y values
Cos x values
Tan sin/
cos
Slide13Steps
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
Slide14Example
Given that cos
θ
= 8/17 and tan
θ
< 0, find all six trig functions.
Triangle is in 4
th
quadrant because that is
where cos is positive and tan is negative
Slide15AssessmentHotseat