Pre-calculus Notes: Chapter 5 The Trigonometric Functions. Use the word bank below to fill in the blanks below. You may use each term only once.

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1 Name: Pre-calculus Notes: Chapter 5 The Trigonometric Functions Section 1 Angles and Degree Measure Use the word bank below to fill in the blanks below. You may use each term only once. degree vertex negative angle terminal side positive angle initial side standard position An angle may be generated by rotating one of two rays that share a fixed endpoint known as a. One of the rays is fixed to form the of the angle, and the second ray rotates to form the. If the rotation is in a counterclockwise direction, the angle formed is a If the rotation is clockwise, it is a. An angle with its vertex at the origin and its initial side along the positive x-axis is said to be in. In the figures below, all the angles are in standard position. The most common unit used to measure angles is the. In order to obtain a more accurate angle measure, Babylonians also measured angles in minutes and seconds. A degree is subdivided into 60 equal parts known as minutes (1 ), and the minute is subdivided into 60 equal parts know as seconds (1 ). Change the angle measure to degrees, minutes, and seconds. a o b o 1

2 Example 2 Rewrite as a decimal rounded to the nearest thousandth. a. 39 o 5 34 b. 35 o 12 7 If the terminal side of an angle that is in standard position coincides with one of the axes, the angle is called a quadrantal angle, as in the figures below. A full rotation around a circle is Measures of more than represent multiple rotations. Example 3 Give the angle measure represented by each rotation. a. 9.5 rotations clockwise b rotations counterclockwise 2

3 Examine the table below to determine the definitions for coterminal and reference angles. It may help to sketch each angle. Coterminal Angles Reference Angles Examples for 60 o Non-Examples for 60 o Example for 60 o Non-Examples for 60 o 420 o 780 o -300 o -660 o 120 o -60 o 30 o 60 o 60 o 120 o 420 o 780 o -300 o Examples for 135 o Non-Examples for 135 o Example for 135 o Non-Examples for 135 o 495 o 855 o -225 o -585 o 135 o 45 o -135 o 225 o 45 o -135 o 225 o -225 o -45 o Examples for 210 o Non-Examples for 210 o Example for 210 o Non-Examples for 210 o 570 o 930 o -150 o -510 o 30 o 210 o 150 o -210 o 30 o -150 o 150 o 210 o -210 o Examples for -20 o Non-Examples for -20 o Example for -20 o Non-Examples for -20 o 340 o 700 o -380 o -740 o 20 o -340 o -20 o 340 o 20 o -20 o 340 o -340 o Coterminal Angles: Reference Angles: Example 4 Identify all angles that are coterminal with each angle. Then find one positive angle and one negative angle that are coterminal with the angle. a. 86 o b. 294 o Example 5 If each angle is in standard position, determine a coterminal angle that is between 0 o and 360 o. State the quadrant in which the terminal side lies. a. 595 o b o 3

4 Example 6 Find the measure of the reference angle for each angle. a b c d Section 2 Trigonometric Ratios in Right Triangles Find the values of sine, cosine, and tangent for A. Example 2 6 a. If sec, find cos. b. If sin 0. 8, find csc. 5 4

5 Example 3 Find the values of the six trigonometric ratios for E. Section 3 Trigonometric Functions on the Unit Circle cos x sin y tan y x sec 1 x csc 1 y cot x y Use the unit circle to find each value. Be cautious: division by zero is undefined, so there are values of tangent, cotangent, secant, and cosecant that are undefined. a. sin(-90 0 ) b. cot c. sec 90 0 d. cos( ) 5

6 Example 2 Use the unit circle to find the values of the six trigonometric functions for a angle. Example 3 Find the values of the six trigonometric functions for angle in standard position if a point with the coordinates (-15, 20) lies on its terminal side. Example 4 Suppose is an angle in standard position whose terminal side lies in Quadrant III. If find the values of the remaining five trigonometric functions of. 4 sin, 5 Example 5 Suppose is an angle in standard position whose terminal side lies in Quadrant IV. If find the values of the remaining five trigonometric functions of. 29 sec, 5 6

7 Section 4 Applying Trigonometric Functions If J = 50 0 and j = 12, find r. Example 2 The chair lift at a ski resort rises at an angle of and attains a vertical height of 1200 feet. a. How far does the chair lift travel up the side of the mountain? b. A film crew in a helicopter records an overhead view of a skier s downhill run from where she gets off the chair lift at the top to where she gets back on the chair lift for her next run. If the helicopter follows a level flight path, what is the length of that path? 7

8 Example 3 A regular hexagon is inscribed in a circle with diameter 26.6 centimeters. Find the apothem of the hexagon. (Apothem = the measure of the line segment from the center of the polygon to the midpoint of one of its sides) Angle of Elevation Angle of Depression Example 4 An observer in the top of a lighthouse determines that the angles of depression to two sailboats directly in line with the lighthouse are and , If the observer is 125 feet above sea level, find the distance between the boats. 8

9 Section 5.5 Solving Right Triangles Solve each equation. a. tan x = 1 b. 1 sin x 2 Example 2 Evaluate each expression. Assume that all angles are in Quadrant I a. cos arccos b. tan cos c. cos arcsin 3 Example 3 If g = 28 and h = 21, find H. Example 4 Many cities place restrictions on the height and placement of skyscrapers in order to protect residents from completely shaded streets. If a 100-foot building casts an 88-foot shadow, what is the angle of elevation of the sun? 9

10 Example 5 Solve each triangle described, given the triangle below. Round to the nearest tenth. a. K = 40 0, k = 26 b. j = 65, l = 55 Section 5.6 The Law of Sines Solve LMN if L = 29 0, M = 112 0, and l = 22. Example 2 A person in a hot-air balloon observes that the angle of depression to a building on the ground is After ascending vertically 500 feet, the person now observes that the angle of depression is How far is the balloonist now from the building? 10

11 Example 3 Find the area of ABC if a = 4.7, c = 12.4, and B =

12 Section 7 The Ambiguous Case for the Law of Sines Determine the number of possible solutions for each triangle. a. A = 30 0, a = 8, b = 10 b. b = 8, c = 10, B = c. A = 63 0, a = 18, b = 25 d. A = 105 0, a = 73, b = 55 Example 2 Find all solutions for each triangle. If no solutions exist, write none. a. A = 98 0, a = 39, b = 22 12

13 b. A = , a = 21, b = 22 Example 3 A group of contractors is constructing a 24-foot slide on a playground. The slide inclines 45 0 from the horizontal. The access ladder measures 18 feet long. At what angle to the horizontal should the contractors build the ladder? 13

14 Section 8 The Law of Cosines Suppose you want to fence a triangular lot. If two sides measure 84 feet and 78 feet and the angle between the two sides is 102 0, what is the length of the fence to the nearest foot? Example 2 Solve each triangle. a. A = , b = 12, c = 14 b. a = 19, b = 24.3, c =

15 Example 3 Find the area of ABC if a = 24, b = 52, and c = 39. Example 4 Find the area of ABC. Round to the nearest tenth. 15