1. WARM UP Sin θ = Cot θ = Cos θ = Sec θ = Tan θ =Csc θ =

3. OBJECTIVES Use function values to find the angles. Be able to find unknown measures in a right triangle. Given two sides of a right triangle or a side and an acute angle, find measures of the other sides and angles.

4. TERMS & CONCEPTS Inverse trigonometric functions Principal branch Mathematical-world answer Real-world answer

5. INVERSES OF TRIGONOMETRIC FUNCTIONS In order to find the measure of an angle when its function value is given, you could press the appropriate inverse function keys on your calculator. For instance: The symbol is the familiar inverse function terminology of Chapter 1, read as “inverse of cosine of 0.8.” Note that it does not mean -1 power of cos, which is the reciprocal of cosine

6. INVERSES OF TRIGONOMETRIC FUNCTIONS Some calculators avoid this difficulty by using the symbol acos x, where “a” can be thought of as standing for “angle.” You can call x or acos x “an angle whose cosine is x.” Trigonometric functions are periodic, so they are not one-to-on functions. There are many angles whose cosine is 0.8. However, for each trigonometric function there is a principal branch of the function that is one-to- one and includes angles between 0° and 90°. The calculator is programmed to give the one angle on the principal branch. The symbol 0.8 means the one angle on the principal branch whose cosine is 0.8.

7. INVERSES OF TRIGONOMETRIC FUNCTIONS The inverse of the cosine function on the principal branch is a function denoted x.

8. DEFINITIONS DEFINITIONS: Inverse Trigonometric Functions If x is the value of a trigonometric function at angle θ, then the inverse trigonometric functions can be defined on limited domains. means sin θ = x and -90° ≤ θ ≤ 90. means cos θ = x and 0° ≤ θ ≤ 180° means tan θ = x and -90° ≤ θ ≤ 90°

9. NOTES Words: “The angle θ is the angle on the principal branch whose sine (and so on) is x” Pronunciation: “Inverse sine of x” and so on, never “sine to the negative 1.” The symbol sin x, cos x, and tan are used only for the value the calculator gives you, not for other angles that have the same function values. The symbols cot x, sec x, and csc x are similarly defined. Caution: The symbol sin does not mean the reciprocal

10. RIGHT TRIANGLE PROBLEMS Trigonometric functions and inverse trigonometric functions often come up in applications, such as right triangle problems. Suppose you have the job of measuring the height of the local water tower. Climbing makes you dizzy, so you decide to do the whole job at ground level. You find that from a point 47.3 m from the base of the tower you must look up at an angle of 53° (angle of elevation) to see the top of the tower. How high is the tower?

11. RIGHT TRIANGLE PROBLEMS Solution: Sketch an appropriate right triangle. Label the known and unknown information h = 47.3 tan 53° = 62.7692… The tower is about 62.8 m high. Note that the angle must always have the degree sign, even during the computations. The symbol tan 53 has a different meaning, as you will learn when you study radians in the next chapter. Write a ratio for tangent Solve for h

12. EXAMPLE 2 A ship is passing through the Strait 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 θ did the ship’s bearing from Gibraltar change? b. How far did the ship travel between the two observations?

13. SOLUTION a. Draw the right triangle and label the unknown angle θ. By the definition cosine, Take the inverse cosine to find θ The angle measure is about 25.09°.

14. SOLUTION b. Label the unknown side, d for distance. By the definition of sine, Use the unrounded angle measure that is in your calculator. The ship traveled about 1124 m. d = 2650 sin 25.0876…° = 1123.6102…

15. Ch. 2.5 Assignments Textbook pg. 83 #2, 4, 6, 8, 9, 16 & 20