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2 Acute Angles and Right Triangles Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 2 Acute Angles and Right Triangles 2.1 Trigonometric Functions of Acute Angles 2.2 Trigonometric Functions of Non-Acute Angles 2.3 Finding Trigonometric Function Values Using a Calculator 2.4 Solving Right Triangles 2.5 Further Applications of Right Triangles Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2 2.1 Trigonometric Functions of Acute Angles Right-Triangle-Based Definitions of the Trigonometric Functions ▪ Cofunctions ▪ Trigonometric Function Values of Special Angles Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 2.1 Example 1 Finding Trigonometric Function Values of An Acute Angle (page 46) Find the sine, cosine, and tangent values for angles D and E in the figure. A C B sin A cos A tan A Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 2.1 Example 1 Finding Trigonometric Function Values of An Acute Angle (cont.) Find the sine, cosine, and tangent values for angles D and E in the figure. A C B sin B cos B tan B Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 2.1 Example 2 Writing Functions in Terms of Cofunctions (page 47) Write each function in terms of its cofunction. (a) sin 9° = cos (90° – 9°) = cos 81° (b) cot 76° = tan (90° – 76°) = tan 14° (c) csc 45° = sec (90° – 45°) = sec 45° Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 2.1 Example 3(a) Solving Equations Using the Cofunction Identities (page 48) Find one solution for the equation. Assume all angles involved are acute angles. (a) Since cotangent and tangent are cofunctions, the equation is true if the sum of the angles is 90º. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 2.1 Example 3(b) Solving Equations Using the Cofunction Identities (page 48) Find one solution for the equation. Assume all angles involved are acute angles. (b) Since secant and cosecant are cofunctions, the equation is true if the sum of the angles is 90º. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 2.1 Example 4 Comparing Function Values of Acute Angles (page 49) Determine whether each statement is true or false. (a) tan 25° < tan 23° In the interval from 0° to 90°, as the angle increases, the tangent of the angle increases. tan 25° < tan 23° is false. (b) csc 44° < csc 40° In the interval from 0° to 90°, as the angle increases, the sine of the angle increases, so the cosecant of the angle decreases. csc 44° < csc 40° is true. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 2.1 Example 5 Finding Trigonometric Values for 30° (page 50) Find the six trigonometric function values for a 30° angle. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 2.1 Example 5 Finding Trigonometric Values for 30° (cont.) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 2.2 Trigonometric Functions of Non-Acute Angles Reference Angles ▪ Special Angles as Reference Angles ▪ Finding Angle Measures with Special Angles Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 2.2 Example 1(a) Finding Reference Angles (page 55) Find the reference angle for 294°. 294 ° lies in quadrant IV. The reference angle is 360° – 294° = 66°. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 2.2 Example 1(b) Finding Reference Angles (page 55) Find the reference angle for 883°. Find a coterminal angle between 0° and 360° by dividing 883° by 360°. The quotient is about 2.5. 883° is coterminal with 163°. The reference angle is 180° – 163° = 17°. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14 2.2 Example 2 Finding Trigonometric Functions of a Quadrant II Angle (page 56) Find the values of the six trigonometric functions for 225°. The reference angle for 225° is 45°. Since 225° lies in quadrant III, the tangent and cotangent are positive while the sine, cosine, secant and cosecant are negative. 2 sin225 sin 45 2 2 cos 225 cos 45 2 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 2.2 Example 2 Finding Trigonometric Functions of a Quadrant II Angle (page 56) 2 cos 225 cos 45 2 tan225 tan 45 1 cot 225 cot 45 1 sec 225 sec 45 2 cscc 225 csc 45 2 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 2.2 Example 3(a) Finding Trigonometric Function Values Using Reference Angles (page 57) Find the exact value of sin(–150°). An angle of –150° is coterminal with an angle of –150° + 360° = 210°. The reference angle is 210° – 180° = 30°. Since an angle of –150° lies in quadrant III, its sine is negative. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 2.2 Example 3(b) Finding Trigonometric Function Values Using Reference Angles (page 57) Find the exact value of cot(780°). An angle of 780° is coterminal with an angle of 780° – 2 ∙ 360° = 60°. The reference angle is 60°. Since an angle of 780° lies in quadrant I, its cotangent is positive. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18 2.2 Example 4 Evaluating an Expression with Function Values of Special Angles (page 57) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19 2.2 Example 5 Using Coterminal Angles to Find Function Values (page 57) Evaluate each function by first expressing the function in terms of an angle between 0° and 360°. (a) (b) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20 2.2 Example 6 Finding Angle Measures Given an Interval and a Function Value (page 58) Find all values of θ, if θ is in the interval [0°, 360°) and sin θ = Since sin θ is negative, θ must lie in quadrants III or IV. The absolute value of sin θ is angle is 60°. so the reference The angle in quadrant III is 60° + 180° = 240°. The angle in quadrant IV is 360° – 60° = 300°. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21 2.3 Finding Trigonometric Functions Using a Calculator Finding Function Values Using a Calculator ▪ Finding Angle Measures Using a Calculator Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22 2.3 Example 1 Finding Function Values with a Calculator (page 62) Approximate the value of each expression. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 23 2.3 Example 1 Finding Function Values with a Calculator (cont.) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24 2.3 Example 2 Using Inverse Trigonometric Functions to Find Angles (page 62) Use a calculator to find an angle θ in the interval [0°, 90°] that satisfies each condition. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 25 2.3 Example 3 Finding Grade Resistance (page 63) The force F in pounds when an automobile travels uphill or downhill on a highway is called grade resistance and is modeled by the equation , where θ is the grade and W is the weight of the automobile. If the automobile is moving uphill, then θ > 0°; if it is moving downhill, then θ < 0°. (a) Calculate F to the nearest 10 pounds for a 5500-lb car traveling an uphill grade with θ = 3.9°. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 26 2.3 Example 3 Finding Grade Resistance (cont.) (b) Calculate F to the nearest 10 pounds for a 2800-lb car traveling a downhill grade with θ = –4.8°. (c) A 2400-lb car traveling uphill has a grade resistance of 288 pounds. What is the angle of the grade? Copyright © 2013, 2009, 2005 Pearson Education, Inc. 27 2.4 Solving Right Triangles Significant Digits ▪ Solving Triangles ▪ Angles of Elevation or Depression Copyright © 2013, 2009, 2005 Pearson Education, Inc. 28 2.4 Example 1 Solving a Right Triangle Given an Angle and a Side (page 69) Solve right triangle ABC, if B = 28°40′ and a = 25.3 cm. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 29 2.4 Example 1 Solving a Right Triangle Given an Angle and a Side (cont.) Three significant digits Copyright © 2013, 2009, 2005 Pearson Education, Inc. 30 2.4 Example 1 Solving a Right Triangle Given an Angle and a Side (cont.) Three significant digits Copyright © 2013, 2009, 2005 Pearson Education, Inc. 31 2.4 Example 1 Solving a Right Triangle Given an Angle and a Side (cont.) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 32 2.4 Example 2 Solving a Right Triangle Given Two Sides (page 70) Solve right triangle ABC, if a = 44.25 cm and b = 55.87 cm. Use the Pythagorean theorem to find c: Four significant digits Copyright © 2013, 2009, 2005 Pearson Education, Inc. 33 2.4 Example 2 Solving a Right Triangle Given Two Sides (cont.) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 34 2.4 Example 2 Solving a Right Triangle Given Two Sides (cont.) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 35 2.4 Example 3 Finding a Length When the Angle of Elevation is Known (page 71) The angle of depression from the top of a tree to a point on the ground 15.5 m from the base of the tree is 60.4°. Find the height of the tree. The measure of equals the measure of the angle of depression because the two angles are alternate interior angles, so Copyright © 2013, 2009, 2005 Pearson Education, Inc. 36 2.4 Example 3 Finding a Length When the Angle of Elevation is Known (cont.) Three significant digits The tree is about 27.3 m tall. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 37 2.4 Example 4 Finding an Angle of Depression (page 72) Find the angle of depression from the top of a 97-ft building to the base of a building across the street located 42 ft away. 97 tan B 42 97 B tan 42 1 B 67 The angle of depression is about 67°. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 38 2.5 Further Applications of Right Triangles Bearing ▪ Further Applications Copyright © 2013, 2009, 2005 Pearson Education, Inc. 39 2.5 Example 1 Solving Problem Involving Bearing (First Method) (page 83) Radar stations A and B are on an east-west line, 8.6 km apart. Station A detects a plane at C, on a bearing of 53°. Stations B simultaneously detects the same plane, on a bearing of 323°. Find the distance from B to C. A line drawn due north is perpendicular to an eastwest line, so right angles are formed at A and B, and angles CAB and CBA can be found as shown in the figure. Angle C is a right angle because angles CAB and CBA are complementary. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 40 2.5 Example 1 Solving Problem Involving Bearing (First Method) (cont.) Find distance a by using the cosine function for angle A. The distance from B to C is about 5.2 km. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 41 2.5 Example 2 Solving Problem Involving Bearing (Second Method) (page 78) A ship leave port and sails on a bearing of S 34° W for 2.5 hr. It then turns and sails on a bearing of N 56° W for 3 hr. If the ships rate of speed is 18 knots (nautical miles per hour), find the distance that the ship is from port. The port is at A. Because the NS lines are parallel, the measure of angle ACD is 34. The measure of angle ACB = 34 + 56 = 90, making ABC a right triangle. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 42 2.5 Example 2 Solving Problem Involving Bearing (Second Method) (page 78) The ship traveled at 18 knots for 2.5 hours or 18(2.5) = 45 nautical miles from A to C. The ship traveled at 18 knots for 3 hours or 18(3) = 54 nautical miles from C to B. 54 45 Use the Pythagorean theorem to find the distance from B to A. a2 b2 c 2 542 452 c 2 4941 c 2 70 c The ship is about 70 nautical miles from port. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 43 2.5 Example 3(a) Using Trigonometry to Measure a Distance (page 79) A method that surveyors use to determine a small distance d between two points P and Q is called the subtense bar method. The subtense bar with length b is centered at Q and situated perpendicular to the line of sight between P and Q. (a) Find d with θ = 2°41′38″ and b = 3.5000 cm. From the figure, we have Copyright © 2013, 2009, 2005 Pearson Education, Inc. 44 2.5 Example 3(a) Using Trigonometry to Measure a Distance (cont.) Convert θ to decimal degrees: Five significant digits Copyright © 2013, 2009, 2005 Pearson Education, Inc. 45 2.5 Example 3(b) Using Trigonometry to Measure a Distance (page 79) (b) Angle θ usually cannot be measured more accurately than to the nearest 1″. How much change would there be in the value of d if θ were measured 1″ larger? Since θ is 1″ larger, θ = 2°41′39″ ≈ 2.694167. The difference is Copyright © 2013, 2009, 2005 Pearson Education, Inc. 46 2.5 Example 4 Solving a Problem Involving Angles of Elevation (page 79) Marla needs to find the height of a building. From a given point on the ground, she finds that the angle of elevation to the top of the building is 74.2°. She then walks back 35 feet. From the second point, the angle of elevation to the top of the building is 51.8°. Find the height of the building. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2-47 47 2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.) There are two unknowns, the distance from the base of the building, x, and the height of the building, h. In triangle ABC In triangle BCD Copyright © 2013, 2009, 2005 Pearson Education, Inc. 48 2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.) Set the two expressions for h equal and solve for x. Since h = x tan 74.2°, substitute the expression for x to find h. The building is about 69 feet tall. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 49 2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.) Graphing Calculator Solution Superimpose coordinate axes on the figure with D at the origin. The coordinates of A are (35, 0). The tangent of the angle between the x-axis and the graph of a line with equation y = mx + b is the slope of the line. For line DB, m = tan 51.8º. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 50 2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.) Since b = 0, the equation of line DB is The equation of line AB is Use the coordinates of A and the point-slope form to find the equation of AB: Copyright © 2013, 2009, 2005 Pearson Education, Inc. 51 2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.) Graph y1 and y2, then find the point of intersection. The y-coordinate gives the height, h. The building is about 69 feet tall. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 52