Warm up
Find x. Round to the
nearest tenth.
A. 14.1
B. 17.4
C. 19.4
D. 21.3
Warm up
Find x. Round to the
nearest tenth.
A. 9.5
B. 15.9
C. 23.7
D. 30.8
Unit 6 Lesson 3
Inverse Trig
Real World Application
• I can use trigonometric ratios to find angle
measures in right triangles.
• I can use trigonometric ratios to solve word
problems.
Inverse Trigonometric Ratios
Remember… Solving
Trigonometric Equations
There are only three possibilities for the placement of the variable ‘x”.
Sin  =
Opp
Hyp
Sin X =
A
A
12 cm
25 cm
x
B
B
12
25
Sin 25
=
= 0.48
0.4226 =
x
12
Sin X =
Sin X
25
X = Sin 1 (0.48)
X = 28.6854
1
x
12
x = (12) (0.4226)
x = 5.04 cm
Sin 
=
A
25
B
C
Sin 25
=
0.4226
=
1
x =
Opp
x
x
12 cm
12 cm
x
C
x
Hyp
C
12
x
12
x
12
0.4226
x = 28.4 cm
5
Use a calculator to find the measure of P to the nearest
tenth.
The measures given are those of the leg adjacent to P
and the hypotenuse, so write the equation using the
cosine ratio.
KEYSTROKES: 2nd [COS] ( 13 ÷ 19 )
Answer:
ENTER
46.82644889
So, the measure of P is approximately 46.8°.
Use a calculator to find the measure of D to the nearest
tenth.
A. 44.1°
B. 48.3°
C. 55.4°
D. 57.2°
Find x. Round to the
nearest tenth.
A. 34.7
B. 43.8
C. 46.2
D. 52.5
Solve the right triangle.
What are we looking for?
 ALL sides of the triangle
 ALL angles of the triangle
Round side measures to the nearest hundredth and angle
measures to the nearest degree.
Step 1
Find mA by using a tangent ratio.
Definition of inverse
tangent
29.7448813 ≈ mA
Use a calculator.
So, the measure of A is about 30.
Step 2
Find mB using complementary angles.
mA + mB = 90
30 + mB ≈ 90
mB ≈ 60
Definition of
complementary
angles
mA ≈ 30
Subtract 30 from
each side.
So, the measure of B is about 60.
Step 3
Find AB by using the Pythagorean Theorem.
(AC)2 + (BC)2 = (AB)2 Pythagorean Theorem
72 + 42
= (AB)2
Substitution
65
= (AB)2
Simplify.
Take the positive
square root of each
side.
Answer:
8.06 ≈ AB
Use a calculator.
mA ≈ 30, mB ≈ 60, AB ≈ 8.06
Solve the right triangle. Round side measures to the
nearest tenth and angle measures to the nearest degree.
A. mA = 36°, mB = 54°,
AB = 13.6
B. mA = 54°, mB = 36°,
AB = 13.6
C. mA = 36°, mB = 54°,
AB = 16.3
D. mA = 54°, mB = 36°,
AB = 16.3
• Angle of elevation
The angle formed by a horizontal line
(usually the ground) and an
observer’s line of sight to an object
above
angle of elevation
• Angle of Depression
The angle formed by a horizontal line
(usually imagined) and an observer’s line
of sight to an object below
angle of depression
Angle of Elevation
CIRCUS ACTS At the circus, a person in the audience at
ground level watches the high-wire routine. A 5-foot-6inch tall acrobat is standing on a platform that is 25 feet
off the ground. How far is the audience member from the
base of the platform, if the angle of elevation from the
audience member’s line of sight to the top of the acrobat
is 27°?
Make a drawing.
Since QR is 25 feet and RS is 5 feet 6 inches or 5.5 feet,
QS is 30.5 feet. Let x represent PQ.
Multiply both sides by x.
Divide both sides by tan
Simplify.
DIVING At a diving competition, a 6-foot-tall diver stands
atop the 32-foot platform. The front edge of the platform
projects 5 feet beyond the ends of the pool. The pool itself is
50 feet in length. A camera is set up at the opposite end of
the pool even with the pool’s edge. If the camera is angled
so that its line of sight extends to the top of the diver’s head,
what is the camera’s angle of elevation to the nearest
degree?
A. 37°
B. 35°
C. 40°
D. 50°
DISTANCE Maria is at the top of a cliff and sees a seal in
the water. If the cliff is 40 feet above the water and the
angle of depression is 52°, what is the horizontal distance
from the seal to the cliff, to the nearest foot?
Make a sketch of the situation.
Since
are parallel,
mBAC = mACD by the
Alternate Interior Angles
Theorem.
Let x represent the horizontal distance from the seal to
the cliff, DC.
c = 52°; AD = 40, and
DC = x
Multiply each side by x.
Divide each side by tan 52°.
The seal is about 31 feet from the cliff.
Luisa is in a hot air balloon 30 feet above the ground. She
sees the landing spot at an angle of depression of 34.
What is the horizontal distance between the hot air
balloon and the landing spot to the nearest foot?
A. 19 ft
B. 20 ft
C. 44 ft
D. 58 ft
DISTANCE Vernon is on the top deck of a cruise ship and
observes two dolphins following each other directly away
from the ship in a straight line. Vernon’s position is 154
meters above sea level, and the angles of depression to
the two dolphins are 35° and 36°. Find the distance
between the two dolphins to the nearest meter.
Understand
ΔMLK and ΔMLJ are right triangles.
The distance between the dolphins is
JK or JL – KL. Use the right triangles
to find these two lengths.
Plan
Because
are horizontal
lines, they are parallel. Thus,
and
because they are alternate interior
angles. This means that
Solve
Multiply each side by JL.
Divide each side by tan
Use a calculator.
Multiply each side by KL.
Divide each side by tan
Use a calculator.
Answer:
The distance between the dolphins is
JK – KL. JL – KL ≈ 219.93 – 211.96,
or about 8 meters.
Madison looks out her second-floor window, which is 15
feet above the ground. She observes two parked cars. One
car is parked along the curb directly in front of her window
and the other car is parked directly across the street from
the first car. The angles of depression of Madison’s line of
sight to the cars are 17° and 31°. Find the distance
between the two cars to the nearest foot.
A. 14 ft
B. 24 ft
C. 37 ft
D. 49 ft