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Using trig ratios in equations
Remember back in 1st grade when you had to
solve:
(6)12 = x (6)
What did you do?
6
72 = x
Remember back in 3rd grade when x was in
the denominator?
(x)12 = 6 (x)
What did you do?
x
__
__
12x = 6
x = 1/2
Three Types Trigonometric Ratios
There are 3 kinds of trigonometric
ratios we learned.
sine ratio
cosine ratio
tangent ratio
Trigonometric Ratios
 When do we use them?
– On right triangles that are NOT 45-45-90 or 30-6090
Find: tan 45
1
Why?
tan = opp
adj
If using a calculator…
I want to find…
Use these calculator keys
Side length measure
SIN
COS
TAN
Angle measure
SIN-1
COS-1
TAN-1
This time, you’re looking for theta.
Ask yourself:
In relation to the angle, what pieces
do I have? Opposite and hypotenuse
42 cm
22 cm
θ
Ask yourself:
What trig ratio uses opposite
and hypotenuse? sine
Set up the equation (remember you’re looking for theta):
Sin θ = 22
42
Remember to use the inverse function
when you find theta
Sin -1 22 = θ
31.59°= θ
42
You’re still looking for theta.
θ
Ask yourself:
22 cm
17 cm
What trig ratio uses the parts I
was given? tangent
Set it up, solve it, tell me what you get.
tan θ = 17
22
tan -1 17 = θ
22
37.69°= θ
Your turn:
In the figure, find sin 
Sin =
=
=
Opposite Side
hypotenuse
4
7
34.85

4
7
 Solve the right
triangle. Round the
decimals to the
nearest tenth.
C
3
B
2
c
A
 Solve the right
triangle. Round
decimals to the
nearest tenth.
sin H =
opp.
hyp.
H
You are looking for
opposite and
hypotenuse which is
the sin ratio.
g
25°
13
J
h
G
Set up the correct ratio
h 13
13 sin 25° =
13
Substitute values/multiply by reciprocal
13(0.4226) ≈ h
Substitute value from table or calculator
5.5 ≈ h
Use your calculator to approximate.
 How do we use right triangles to solve
real life problems?
Indirect Measurement
Find the height of the tree given an angle and the
length of the shadow of the tree.
opposite
tan 59 =
adjacent
h
h
tan 59 =
45
45 tan 59 = h
74.9  h
59
45 ft
Using Right Triangles
in Real Life
 Space Shuttle: During its
approach to Earth, the
space shuttle’s glide angle
changes.
 When the shuttle’s altitude
is about 15.7 miles, its
horizontal distance to the
runway is about 59 miles.
What is its glide angle?
Round your answer to the
nearest tenth.
Solution:
 You know opposite
and adjacent sides.
If you take the
opposite and divide it
by the adjacent side,
then take the inverse
tangent of the ratio,
this will give you the
slide angle.
15.7
miles
x°
59 miles
opp.
Use correct ratio
tan x° =
adj.
15.7
Substitute values
tan x° =
59
≈ 14.9
 When the space shuttle’s altitude is about 15.7 miles, the
glide angle is about 14.9°.
 When the space shuttle is
5 miles from the runway,
its glide angle is about 19°.
Find the shuttle’s altitude
at this point in its descent.
Round your answer to the
nearest tenth.
tan 19° =
tan 19° =
opp.
adj.
h
5 tan 19° =
5
19°
5 miles
Use correct ratio
Substitute values
5
h
h
5
Isolate h by
multiplying by 5.
1.7 ≈ h Approximate using calculator
Conclusion
opposite side
sin  
hypotenuse
adjacent side
cos  
hypotenuse
opposite side
tan  
adjacent side
Make Sure
that the
triangle is
right-angled