Download SOH CAH TOA (angles)

Applications of Trig Functions
•  Today we’ll be looking at ways to solve realworld situations using the trig functions we
learned last time.
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Sometimes we know the length of the sides of a
triangle but need to know the angles.
Guess what? We can use trig functions to do this!
7
x°
12
Get out your calculators we’re
going to learn about Trig
Inverses! The way of “undo-ing”
trig functions!
If you know the sine, cosine, or tangent of an acute
angle measure, you can use the inverse
trigonometric functions to find the measure of the
angle.
Use your calculator to find each angle measure
to the nearest degree.
A. cos-1(0.87)
B. sin-1(0.85)
C. tan-1(0.71)
cos-1(0.87) ≈ 30°
sin-1(0.85) ≈ 58°
tan-1(0.71) ≈ 35°
Find m∠A.
hypotenuse
A
19
We should use SOH
13
SinA = ≈ .6842
19
SinA ≈ .6842
Sin −1SinA ≈ Sin −1 (.6842)
€
€
C
13
B
opposite
Find m∠B.
m∠B = 90° − m∠A
A ≈ 43.17°
m∠B = 90° − 43.17° = 46.83°
€
Find m∠B.
hypotenuse
A
23
We should use CAH
18
CosB =
≈ .7826
23
CosB ≈ .7826
Cos−1CosB ≈ Cos−1 (.7826)
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€
C
18
B
adjacent
Find m∠A.
m∠A = 90° − m∠B
B ≈ 38.5°
m∠A = 90° − 38.5° = 51.5°
€
Find m∠B.
A
We should use TOA
€
18
B
adjacent
TanB ≈ .5
Tan −1TanB ≈ Tan −1 (.5)
€
9
C
9
TanB = ≈ .5
18
€
opposite
Find m∠B.
m∠B = 90° − m∠A
B ≈ 26.57°
m∠B = 90° − 26.57° = 63.43°
€
Using given measures to find the unknown angle
measures or side lengths of a triangle is known as
solving a triangle.
To solve a right triangle, you need to know two side
lengths or one side length and an acute angle
measure.
Remember that you can always use the Pythagorean
Theorem too.
Find ST, m∠R, and m∠T.
RT2 = RS2 + ST2
(5.7)2 = 52 + ST2
Since the acute angles of a right triangle are
complementary, m∠T ≈ 90° – 29° ≈ 61°.
Find m∠D, EF, and DF
Baldwin St. in Dunedin, New Zealand, is the
steepest street in the world. It has a grade of
38%. To the nearest degree, what angle does
Baldwin St. make with a horizontal line?
A 38% grade means the road rises (or falls) 38 ft for
every 100 ft of horizontal distance.
C
38 ft
A
100 ft
B
An angle of elevation is the angle formed by a
horizontal line and a line of sight to a point above
the line. In the diagram, ∠1 is the angle of elevation
from the tower T to the plane P.
An angle of depression is the angle formed by a
horizontal line and a line of sight to a point below
the line. ∠2 is the angle of depression from the
plane to the tower.
Use the diagram above to
classify each angle as an angle
of elevation or angle of
depression.
The Seattle Space Needle casts a
67-meter shadow. If the angle of
elevation from the tip of the
shadow to the top of the Space
Needle is 70º, how tall is the
Space Needle? Round to the
nearest meter.
Suppose a forest ranger sees a fire and measures
the angle of depression to the fire is 3°. What is the
horizontal distance to this fire? Round to the
nearest foot.
3°
An observer in a lighthouse is 69 ft above the
water. He sights two boats in the water directly
in front of him. The angle of depression to the
nearest boat is 48º. The angle of depression to
the other boat is 22º. What is the distance (x)
between the two boats? Round to the nearest
foot.
A pilot flying at an altitude of 12,000 ft sights
two towns directly in front of him. The angle of
depression to one town is 78°, and the angle
of depression to the second town is 19°. What
is the distance (x) between the two towns?
Round to the nearest foot.
19°
78°
12,000 ft
78°
19°