Chapter 2 Section 5

Inverse trigonometric Functions and Triangle Problems


Objective: given two sides of a right triangle or a side and an acute angle, find measures of the other sides and angles.


Inverses of Trigonometric Functions
To find the measure of an angle when its function value is given, you could press a keys in your calculator, for example, if cos θ = 0.8, then you would press cos-1 0.8 = 36. 8698


For the symbol cos-1, you would say “inverse of cosine 0.8.”
Note: it DOES NOT mean the -1 power of cos, which is the reciprocal.


The domain of the inverse is the range of the original function and the range for the inverse is the domain of the original.


Trigonometric functions are periodic. They are not one-to-one functions.
Principal Branch: for each trigonometric function there is a principal branch of the function that is one-to-one and includes angles between 0º and 90º. It includes the entire range of original function.


Definitions of the inverse trigonometric function:
If x is the value of a trigonometric function at angle θ, then the inverse trigonometric functions can be defined on limited domains.

θ = sin-1 x means sin θ =x and -90º < θ <>

θ = cos-1 x means cos θ =x and 0º < θ <>

θ = tan-1 x means tan θ =x and -90º < θ <>

Domain: x <90

Range : all real numbers

Right Triangle Problems

Example:

A ship is passing through the straight of Gibraltar. At its closest point of approach, Gibraltar radar determines that the ship is 2400 m away. Later, the radar determines that the ship is 2650 m away.
a. By what angle did the ship’s bearing from Gibraltar change?
b. How far did the ship travel between the two observations?

Solution:

a. Draw the right triangle and label the unknown angle θ.

By the definitions of cosine,

cos θ = adjacent/hypotenuse= 2400/2650

θ = cos-1 2400/2650 = 25.0876…º take the inverse cosine to find θ

The angle θ is about 25.09º

b. Label the unknown side d, for distance. By the definition of sine,

d/2650 = sin 25.0876…º

d = 2650 sin 250876...º = 1123.6102…

The ship traveled about 1124 m.

want to learn more about Inverse Trigonometric Funtions? go here: http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/zoo/invtrig.html

I’ve always been known to ask stupid/weird questions (I’m sure Debby has heard this before): Do fetuses dream?

Anna, You're up next!!! Have fun :)