Inverse Trigonometry – using information about the sides of right
triangles to find an ANGLE measure.
B
1
Inverse Tangent If tan A  x then tan x  mA
1
Inverse Sine If sin A  y then sin y  mA
1
Inverse Cosine If cos A  z then cos z  mA
C
Finding the measure of an acute angle:
1.) Find the trigonometric ratio between two given sides.
2.) Take the inverse of the trigonometric ratio:
sin  sin-1
cos  cos-1
A
calc mode
to DEGREE
tan  tan-1
Example 1: Find the value of x in the triangle. Round your answer to the nearest degree.
Example 2: Find the value of x in the triangle. Round your answer to the nearest degree.
In word problems, the formulas remain the same:
sin A =
Opposite leg
Hypotenuse
cos A =
Adjacent leg
Hypotenuse
tan A =
Opposite leg
Adjacent leg
Word problems introduce two new vocabulary terms:
Angle of Elevation
Think an elevator
that only goes up!
- always measured from the ground up
- always INSIDE the triangle
- movement of your eyes; you are looking straight
ahead and you must raise (elevate) your eyes to see
the top of a tree or building
Angle of Depression
- always OUTSIDE the triangle
- movement of your eyes; you are standing at the
top of a lighthouse and looking straight ahead, you
must lower (depress) your eyes to see the boat
As seen in the diagram above of angle of depression, the dark horizontal line is
parallel to side CA of triangle ABC. This forms alternate interior angles,
which are congruent. SO… x equals the m Ð BAC!!!! This means:
the angle of elevation = the angle of
depression
Example 3: From a point on the ground
25 feet from the foot of a tree, the angle
of elevation of the top of the tree is 32˚.
Find, to the nearest foot, the height of
the tree.
Example 4: Form the top of a barn 25 feet
tall, you see a cat on the ground. The angle
of depression of the cat is 40˚. How many
feet, to the nearest foot, must the cat walk
to reach the barn?
Example 5: A ladder leaning against a
house makes an angle of 60˚ with the
ground. The foot of the ladder is 7 feet
from the foundation of the house.
How long is the ladder?
Example 6: A balloon on a 40-foot string
makes an angle of depression of 50˚ to the
person holding the balloon. About how
high, to the nearest tenth of a foot, above
the ground is the balloon if the hand of the
person holding the balloon is 6 feet above
the ground?
Example 7:
A child 90 cm high casts a shadow that is 180 cm long. Find the angle of elevation to the sun,
to the nearest degree.
Example 8: You are watching a fireworks display from, x, feet behind the launch pad. The
launch tubes are aimed directly away from you at an angle of 65o with the ground. The angle
of elevation for you to see the fireworks is 40o. The height of the fireworks when they ignite is
400 feet. Find how far away from the launch pad you are standing? Round your final answer to
the nearest foot.
400 ft
x