1. When problem solving, you may be asked to find a missing side of a right triangle. You also may be asked to find a missing angle.
If you look at your calculator, you should be able to find the inverse trig functions. These can be used to find the measure of an angle that has a specific sine, cosine, or tangent.

2. Inverse Trig Functions
What if you know all the sides of a right triangle but you don’t know the other 2 angle measures. How could you find these angle measures?
What you need is the inverse trigonometric functions.
Think of the angle measure as a present. When you take the sine, cosine, or tangent of that angle, it is similar to wrapping your present.
The inverse trig functions give you the ability to unwrap your present and to find the value of the angle in question.

3. Notation A=sin-1(z) is read as the inverse sine of A.
Never ever think of the -1 as an exponent. It may look like an exponent and thus you might think it is 1/sin(z), this is not true.
(We refer to 1/sin(z) as the cosecant of z)
A=cos-1(z) is read as the inverse cosine of A.
A=tan-1(z) is read as the inverse tangent of A.

4. Inverse Trig definitions
Referring to the right triangle from the introduction slide. The inverse trig functions are defined as follows:
A=sin-1(opp/hyp)
A=cos-1(adj/hyp)
A=tan-1(opp/adj)

5. Example using inverse trig functions
Find the angles A and B given the following right triangle.
Find angle A. Use an inverse trig function to find A. For instance A=sin-1(6/10)=36.9°.
Then B = 180° - 90° ° = 53.1°. B A

6. If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle.

7. Example 2: Calculating Angle Measures from Trigonometric Ratios
Use your calculator to find each angle measure to the nearest degree.
A. cos-1(0.87)
B. sin-1(0.85)
C. tan-1(0.71)
cos-1(0.87)  30°
sin-1(0.85)  58°
tan-1(0.71)  35°

8. Using given measures to find the unknown angle measures or side lengths of a triangle is known as solving a triangle. To solve a right triangle, you need to know two side lengths or one side length and an acute angle measure.
Caution!
Do not round until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator.

9. Example 3: Solving Right Triangles
Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.
Method: By the Pythagorean Theorem,
RT2 = RS2 + ST2
(5.7)2 = 52 + ST2
Since the acute angles of a right triangle are complementary, mT  90° – 29°  61°.

10. Check It Out! Example 3
Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.
Since the acute angles of a right triangle are complementary, mD = 90° – 58° = 32°.
, so EF = 14 tan 32°. EF  8.75
DF2 = ED2 + EF2
DF2 =
DF  16.51

11. Special triangles 30 – 60 – 90 degree triangle.
Consider an equilateral triangle with side lengths 2. Recall the measure of each angle is 60°. Chopping the triangle in half gives the 30 – 60 – 90 degree traingle.
30°
√3 2 °

12. 30° – 60° – 90°
Now we can define the sine cosine and tangent of 30° and 60°.
sin(60°)=√3 / 2; cos(60°) = ½; tan(60°) = √3
sin(30°) = ½ ; cos(30°) = √3 / 2; tan(30°) = 1/√3

13. 45° – 45° – 90°
Consider a right triangle in which the lengths of each leg are 1. This implies the hypotenuse is √2.
45° sin(45°) = 1/√2
√2 cos(45°) = 1/√2
1 tan(45°) = 1 °

14. Example Find the missing side lengths and angles.
sin(60°)=y/10
x thus y=10sin(60°)
A y