Trigonometric Ratios II

Objectives:

1. To find missing angles in a right triangle using inverse trigonometric ratios

2. To complete and use the unit circle to find the exact values of various angle measures

Inverse Trigonometry

To find an angle measurement in a right triangle given any two sides, use the inverse of the trig ratio, but each of them are only defined on certain intervals.

Inverse Trigonometry

.tan then ,tan If

.cos then , cos If

.sin then ,sin If

1

1

1

b

ax

b

ax

b

ax

b

ax

b

ax

b

ax 90 90 ; 1 1

0 180 ; 1 1

90 90 ;

ax

ba

xb

ax

b

Inverse Trigonometry

• “sin-1 x” is read “the angle whose sine is x” or “inverse sine of x”

• arcsin x is the same thing as sin-1 x

.tan then ,tan If

.cos then , cos If

.sin then ,sin If

1

1

1

b

ax

b

ax

b

ax

b

ax

b

ax

b

ax

Example 5

Let <A and <B be acute angles in a right triangle. Use a calculator to approximate the measures of <A and <B to the nearest tenth of a degree.

1. sin A = 0.87

2. cos B = 0.15

Type in your calculator as:1sin (0.87)

Example 6

If the legs of a right triangle are 3 and 4, what is the measure of the angle opposite the smallest side?

3

4

Example 7

Find the measures of the acute angles of a 8-15-17 right triangle.

Example 8

Suppose your school is building a raked stage. The stage will be 30 feet long from front to back, with a total rise of 2 feet. A rake (angle of elevation) of 5° or less is generally preferred for the safety and comfort of the actors. Is the raked stage you are building within this suggested range?

Example 9

To solve a right triangle means to find all of its sides and angles. Using trigonometry, what must you know to solve a right triangle?

Example 10

Solve the right triangle. Round your answers to the nearest tenth.

42

70 ft

Example 11

Solve each right triangle. Write your answers in simplest radical form.

30

1 unit

1 unit

45

Radians

Radians

Radians are another way to measure an angle. If you take the radius and wrap it around the circle, the angle that is formed is one radian.

Radians

It takes a little bit more than 3 radians to span a semicircle.

That “little bit more than 3” is π.

So π radians = 180° and 2π radians = 360°

Example 10

Rewrite each of the following angle measures in terms of radians. (180° = π rad)

1. 30°

2. 45°

3. 60°

4. 90°

The Unit Circle

This tiny circle is called the unit circle since its radius is 1 unit. This circle may be tiny, but it will give us a way to find 102 exact trig values. That’s pretty useful.