Algebra 2 Honors
Section 12.1 Notes: Trigonometric Functions in Right Triangles
A trigonometric ratio compares the side lengths of a right triangle. A trigonometric function has a rule given by a trigonometric
ratio.
The Greek letter theta θ is often used to represent the measure of an acute angle in a right triangle. The hypotenuse, the leg opposite θ,
and the leg adjacent to θ are used to define the six trigonometric functions.
sin A 
opposite
adjacent
opposite
, cos A 
, tan A 
hypotenuse
hypotenuse
adjacent
opposite leg
If you recall from Geometry, we learned about three of the six trigonometric ratios (seen below) and the Pythagorean Theorem:
a2+ b2 = c2
A
SOH CAH TOA
θ
C
adjacent
B
There are 3 more trigonometric functions to know!
Example 1: Find the values of the six trigonometric functions for angle G.
Notice that the cosecant, secant, and cotangent ratios are reciprocals of the sine, cosine, and tangent ratios, respectively. These are
called the reciprocal functions.
5
Example 2: If tan A  , find the exact values of the five remaining trigonometric functions for A.
3
B
A
Example 3: If sin B 
2
, find the exact values of the five remaining trigonometric functions for B.
3
B
A
Example 4: Use a trigonometric function to find the value of x.
Example 5: Use a trigonometric function to find the value of x.
30°
x
5
Example 6: Use a trigonometric function to find the value of x.
7
x
26°
Example 7: Use a trigonometric function to find the value of x.
x
47°
12
Example 8:
a) Find the measure of B. Round to the nearest tenth if necessary.
b) Find the measure of A. Round to the nearest tenth if necessary.
Example 9: Find the measure of A. Round to the nearest tenth if necessary.
B
Example 10: Solve ΔABC.
14
A
Example 11: Solve the triangle: If Q is a right angle, R=37⁰, and q=22
b
6
C
Example 11: To calculate the height of a tree in his front yard, Anand walked 50 feet from the base of the tree and used an
inclinometer to measure the angle from his eye to the top of the tree, which was 62°. If Anand’s eye level is at 6 feet, about how tall
is the tree?
Example 12: To calculate the height of a building, Joel walked 200 feet from the base of the building and used an inclinometer to
measure the angle from his eye to the top of the building. If Joel’s eye level is at 6 feet, what is the distance from the top of the
building to Joel’s eye?
Example 13: A golfer is standing at the tee, looking up to the green on a hill. The tee is 36 yards lower than the green and the angle of
elevation from the tee to the hole is 12°. From a camera in a blimp, the apparent distance between the golfer and the hole is the
horizontal distance. Find the horizontal distance.
Example 14: The hill of the roller coaster has an angle of descent, or an angle of depression, of 60°. Its vertical drop is 195 feet. From
a blimp, the apparent distance traveled by the roller coaster is the horizontal distance from the top of the hill to the bottom. Find the
horizontal distance.