Chapter 5: The Trigonometric
Functions
Section 5-5: Solving Right Triangles
Objectives
• Evaluate inverse trigonometric functions
• Find missing angle measurements
• Solve right triangles
Review unit circle
•
In lesson 5-3, you learned to use the unit circle to determine the value of trigonometric
functions. Turn to page 305 in your book. These are the frequently-used values.
Inverse of the trigonometric functions
•
If you know the trigonometric value and need to find the angle, you use the
inverse of the function. The inverse of the sine function is the arcsine function.
The inverse of the cosine function is the arccosine function.
•
An equation such as sin x = ½ can be written as x = arcsine 1/2
Inverse function and the ordered pair
The inverse function may be found by interchanging the coordinates of the
ordered pairs of the function. The domain of the function becomes the
range of its inverse and the range of the function becomes the domain of
its inverse.
The inverse of sine is the set of ordered pairs (sin x , x) The domain of the
inverse sine is -1≤x≤1
Example: Solve tan x = 1
If tan x = 1, then x is an angle whose tangent is 1.
X=arctan 1 Using your graphing calculator, you can determine that x
equals 45 degrees, 225 degrees or any angles coterminal with these
angles.
-1
Evaluate tan (Cos 4/5)
-1
Let A= Cos 4/5
Then Cos A = 4/5 by the definition of inverse
Now if we draw a diagram of angle A in Quadrant 1, we have
5
A
Use the Pythagorean Theorem and we
find the missing side is 3.
4
Since tangent is side opposite / side adjacent, we have 3/4
Inverse trigonometric relations can be used to find the measure of
angles of right triangles. Calculators can be used to find the values of the
inverse trigonometric relations. Make sure you are in degree mode on
your calculator! Also the ration will need to be enclosed in parentheses
to correctly evaluate the inverse function.
Example:
If g = 28 and h= 21, find angle H.
H
28
G
sin H = 21/28
arcsin sin H = arcsin (21/28)
H= 48.6 degrees
F
21
You can use trigonometric functions and inverse
relations to solve right triangles. To solve a triangle
means to find all of the measures of its sides and
angles. Usually, two measures are given. Then you
can find the remaining measures.
Remember:
Inverse functions are
used to find the angles and trigonometric
functions are used to find the ratios of sides.