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Trigonometry – Exact Value, Laws, and Vectors Conversions Arc length and Area of a Sector Arc length: s = θ r ⎛ 180 ⎞ ⎟ ⎝ π ⎠ ⎛• π ⎞ ⎜ ⎟ ⎝ 180 ⎠ rad → deg ⎜ • deg → rad 1 2 θr 2 Only when the angle is in RADIANS! Unit Circle Area of a sector: Area = Coterminal Angles θ + 360D n , n ∈ Ζ Reference Angles ( 0,1) sin A 0 π π ≤y≤ 2 2 π π − <y< 2 2 − Area of a Triangle 1 K = ab sin C 2 a 3 3 0≤y≤π Arc tan θ =T an −1 θ Law of Sines ⎛1 3⎞ 1 ⎜ , ⎟ 3 ⎜⎝ 2 2 ⎟⎠ π ⎛ 2 2⎞ ⎜ ⎟ 45D, , 4 ⎜⎝ 2 2 ⎟⎠ π ⎛ 3 1⎞ ⎜ 30D, , ⎟ 6 ⎜⎝ 2 2 ⎟⎠ π Circle with radius, r ( r cos θ , r sin θ ) Inverse Trigonometry Arc sin θ = S in −1 θ 60D, (1, 0 ) Degree-Minutes-Seconds 2nd APPS (Angle) ENTER 4: DMS Arc cos θ = C os −1 θ 3 undefined = b sin B [SAS] = c sin C [AAS, ASA, *SSA] *used to find # of triangles that can be formed (0, 1, or 2) Law of Cosines a 2 = b 2 + c 2 − 2bc cos A [SAS, SSS] -----------------------------------------------Vector Problems construct a parallelogram and remember properties of parallelograms - opposite sides congruent - opposite angles congruent - consecutive angles supplementary - diagonal creates alternate interior angles which are congruent - only the diagonals of a square or rhombus bisect the angle Conversions and Exact Value 1. What is the radian measure of the smaller angle formed by the hands of a clock at 7 o’clock? π 2π 5π 7π (1) (2) (3) (4) 2 3 6 6 2. Express 405º in radian measure. 3. Find, to the nearest tenth of a degree, the angle whose measure is 2.5 radians. 4. If sin A < 0 and cot A > 0 , in which quadrant does the terminal side of ∠A lie? 5. Find the exact value of each of the following. ( sin 240D 6. ) ( cos 315D ) ( tan 150D ) csc ( 4π ) If f ( x ) = sin x + cos2x , then f (π ) is (1) 1 (2) 2 (3) 0 (4) −1 7. ⎛π ⎞ If f ( x ) = sin2 x + cos2 x , find f ⎜ ⎟ . ⎝4⎠ 8. The accompanying diagram shows unit circle O, with radius OB = 1 . Which line segment has a length equivalent to cos θ ? (1) AB (2) CD (3) OC (4) OA 9. If is an angle in standard position and its terminal side passes through the point , find the exact value of . 10. Express 11. Express as a function of a positive acute angle. as a function of a positive acute angle. 12 Evaluate each of the following: ⎛− 2⎞ a. Cos −1 ⎜ ⎟= ⎜ 2 ⎟ ⎝ ⎠ d. sin (Cos −1 ( −1 ) ) = ⎛ −1 ⎞ b. Sin −1 ⎜ ⎟ = ⎝2⎠ ⎛ 3⎞ c. T an −1 ⎜ − ⎟= ⎜ 3 ⎟ ⎝ ⎠ 7 ⎞ ⎛ e. cos ⎜ Arc sin ⎟ 25 ⎠ ⎝ Arc Length / Area of a Sector 13. An arc of length 30 inches is drawn in a circle with radius 12 inches. (a) Find, to the nearest integer, the degree measure of the arc. (b) Find the area of the sector formed by this arc. 14. A sprinkler system is set up to water the sector shown in the accompanying diagram, with angle ABC measuring 1 radian and radius AB = 20 feet. What is the length of arc AC, in feet? (1) 63 (2) 31 (3) 20 15. (4) 10 The accompanying diagram shows the path of a cart traveling on a circular track of radius 2.40 meters. The cart starts at point A and stops at point B, moving in a counterclockwise direction. What is the length of minor arc AB, over which the cart traveled, to the nearest tenth of a meter? 16. 17. In a circle with a radius of 4 centimeters, what is the number of radians in a central angle that intercepts an arc of 24 centimeters? The pendulum of a clock swings through an angle of 2.5 radians as its tip travels through an arc of 50 centimeters. Find the length of the pendulum, in centimeters. Area & Laws 18. The three sides of a triangle have lengths 23, 25, and 40. (a) Find, to the nearest degree, the measure of the largest angle of the triangle. (b) Find, to the nearest integer, the area of the triangle. 19. In ΔKLM , KL = 100, m ∠K = 40, and LM = 80 . Explain why there are two possible triangles with these measures. 20. In ΔABC , m ∠A = 30, a = 12, and b = 10. Explain why there is only one possible triangle with these measures. 21. In ΔABC , m ∠A = 75, m ∠B = 40 and b = 35 . What is the measure of side c? (1) (3) 35 sin 40D sin 65D 35 sin 40D sin 75D (2) (4) 35 sin 75D sin 40D 35 sin 65D sin 40D 4 3 , sin B = , and a = 16 . Find b. 5 4 22. In triangle ABC, sin A = 23. In ΔABC , m ∠A = 30, a = 12, and b = 10. Which type of triangle is ΔABC ? (1) acute 24. (2) isosceles (3) obtuse (4) right In the accompanying diagram of triangle RST, m ∠R = 17 D20', RT = 40, and m ∠T = 34D50' . What is the length of RS to the nearest integer? 25. The accompanying diagram shows the plans for a cell-phone tower that is to be built near a busy highway. Find the height of the tower, to the nearest foot. 26. A ship at sea heads directly toward a cliff on the shoreline. The accompanying diagram shows the top of the cliff, D, sighted from two locations, A and B, separated by distance S. If m ∠DAC = 30, m ∠DBC = 45 and S = 30 feet, what is the height of the cliff, to the nearest foot? 27. An airplane traveling at a level altitude of 2050 feet sights the top of a 50-foot tower at an angle of depression of 28 D from point A. After continuing in level flight to point B, the angle of depression to the same tower is 34 D . Find, to the nearest foot, the distance that the plane traveled from point A to point B. 28. To the nearest degree, what is the measure of the largest angle in a triangle with sides measuring 10, 12, and 18 centimeters? (1) 109 (2) 81 (3) 71 (4) 32 29. Peter (P) and Jamie (J) have computer factories that are 132 miles apart. They both ship their completed computer parts to Diane (D). Diane is 72 miles from Peter and 84 miles from Jamie. Using points D, J, and P to form a triangle, find m ∠PDJ to the nearest ten minutes or nearest tenth of a degree. Vectors 30. Two forces of 40 pounds and 20 pounds, respectively, act simultaneously on an object. The angle between the two forces is 40°. (a) Find the magnitude of the resultant, to the nearest tenth of a pound. (b) Find the measure of the angle, to the nearest degree, between the resultant and the larger force. 31. Two forces act on a body to produce a resultant force of 70 pounds. One of the forces is 50 pounds and forms an angle of 67 D 40' with the resultant force. Find, to the nearest pound, the magnitude of the other force. 32. One force of 20 pounds and one force of 15 pounds act on a body at the same point so that the resultant force is 19 pounds. Find, to the nearest degree, the angle between the two original forces. 33. A jet is flying at a speed of 526 miles per hour. The pilot encounters turbulence due to a 50-mile-per-hour wind blowing at an angle of 47°, as shown in the accompanying diagram. ` Find the resultant speed of the jet, to the nearest tenth of a mile per hour. Use this answer to find the measure of the angle between the resultant force and the wind vector, to the nearest tenth of a degree. Answers – Trig Day 1 1. (3) 5. 5π 6 3 2 , , 2 2 3 − , undefined 3 9. − 13 2 13. (a) 143D (b) 180 sq. in. 17. 20 cm 21. (4) 25. 88 feet 29. 115D20' or 115.4D 33. 561.3 mph 43.3D 2. 9π 4 6. (1) 1 3. 143.2D 4. Quadrant III 7. 1 8. 10. tan ( 50 ) 11. − sin (10 ) 14. (3) 20 15. 6.9 meters 12. 135D , − 30D , 24 −30D , 0, 25 16. 6 radians 18. (a) 113D (b) 265 units sq. 19. Using Law of Sines there are 2 possible angles and both are possible. 23. (3) obtuse 27. 796 feet 31. 69 pounds 20. Using Law of Sines there are 2 possible angles, but only one is possible. 24. 29 28. (1) 109 32. 116D 22. b = 15 26. 41 feet 30. (a) 56.8 pounds (b) 13D (4) OA