MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Transcription

1 Trigonometry Final Exam Study Guide Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a polar equation is given. Select the polar equation for the graph. 1) 1) r A) r = 5 + cos(5θ) B) r = 5 cos(5θ) C)r = 5 D) r = 5 sin(5θ) Graph the polar equation for θ in [0, 360 ). ) r = + sin θ ) r A) B) r r

2 Trigonometry Final Exam Study Guide C) D) r r Find an equivalent equation in rectangular coordinates. 3) r = cos θ A) x + y = x B) (x + y) = y C)x + y = y D) (x + y) = x 3) Plot the point. ), -5π ) A) -5 B)

3 Trigonometry Final Exam Study Guide C) D) Give the rectangular coordinates for the point. 5) (-6, 5 ) A) (-3, -3 ) B) (3π, 3π) C)(3 3, 3 3) D) (3, 3 ) 5) 6) 3, π 3 6) A) - 3, 3 B) 3, - 3 C) - 3, 3 3 D) 3 3, - 3 The rectangular coordinates of a point are given. Express the point in polar coordinates with r 0 and 0 θ < ) (3, -3) 7) A) (3, 315 ) B) (3, 5 ) C)(3, 135 ) D) (3, 5 ) Find all solutions of the equation. Leave answers in trigonometric form. 8) x = 0 A) { cis 180, cis 10, cis 300 } B) { cis 180, cis 10, cis 0 } C){ cis 180, cis 60, cis 300 } D) { cis 180, cis 0, cis 0 } 8) 9) x 3 + 8i = 0 A) { cis 70, cis 10, cis 150 } B) { cis 70, cis 30, cis 150 } C){ cis 90, cis 150, cis 30 } D) { cis 90, cis 10, cis 330 } 9) Find all cube roots of the complex number. Leave answers in trigonometric form. 10) 15(cos 55 + i sin 55 ) A) 5 cis 85, 5 cis 15, 5 cis 5 B) 5 cis 85, 5 cis 05, 5 cis 05 C)5 cis 175, 5 cis 35, 5 cis 115 D) 5 cis 85, 5 cis 05, 5 cis 35 10) 11) 6 A) cis 5, cis 165, cis 35 B) cis 0, cis 60, cis 10 C) cis 90, cis 10, cis 330 D) cis 0, cis 10, cis 0 11) 3

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5 MA10 Trigonometry Final Exam Study Guide Page 1 Name: Show your work. Give exact answers unless otherwise instructed. Circle your answers. 1) (5 pts) If tan x = 3 with 0 < x < π, find the exact value of tan (x + π). ) (5 pts) If tan x = 3 with 0 < x < π, find the exact value of cot x. 3) (5 pts) If tan x = 3 with 0 < x < π, find the exact value of sin x.

6 MA10 Trigonometry Final Exam Study Guide Page ) (5 pts) Solve the oblique triangle. Round to decimal places if decimal does not terminate in fewer. If there are two solutions, give both of them. If there is only one solution, cross out the table marked Second solution. If there are no solutions, cross out both tables. First solution (if it exists) α β γ 5 a b c 15 cm 33 cm Second solution (if it exists) α β γ 5 a b c 15 cm 33 cm B β c a A α b γ C 5) (5 pts) Given sin x = 1 3 with π < x < π. Find the exact value of sin x. 6) (5 pts) Find all exact solutions of the equation sec x + = 0 on the interval [0, π).

7 MA10 Trigonometry Final Exam Study Guide Page 3 7) (5 pts) A supporting cable runs from the top of a 5-foot antenna to a point on level ground 0 feet from the base of the antenna. What is the angle between the cable and the ground? Round to 1 decimal place if decimal does not terminate in fewer. 8) (5 pts) In the diagram below, the circle with central angle θ has radius 3 cm. The dimensions of the right triangle are x = cm wide and y = cm tall. Answer the following questions. (a) Find the exact value of sin θ. (b) Find the length of the arc cut off by the angle θ. Round to 3 decimal places if decimal does not terminate in fewer. 9) (5 pts) Suppose sin x = 5 13 with 3π < x < π and cos y = 3 5 with π < y < 3π. Find the exact value of sin(x + y).

8 MA10 Trigonometry Final Exam Study Guide Page 10) (5 pts) Solve the oblique triangle. Round to decimal places if decimal does not terminate in fewer. If there are two solutions, give both of them. If there is only one solution, cross out the table marked Second solution. If there are no solutions, cross out both tables. First solution (if it exists) α β γ a b c cm Second solution (if it exists) α β γ a b c cm B β c a A α b γ C 11) (5 pts) Use identities to simplify: form below. (a) sin x (b) cos x (c) tan x (d) cot x (e) sec x (f) csc x csc x sin x. Select the appropriate simplified cos x 1) (5 pts) Given tan x = 3 with π < x < π. Find the exact value of cos x.

9 MA10 Trigonometry Final Exam Study Guide Page 5 13) (5 pts) To find the length AB of a small lake, a surveyor at point C measures ACB to be 115, length AC to be 500 feet, and length BC to be 35 feet. What is the length of the lake (to the nearest foot)? Circle your answer. 1) (5 pts) Find the amplitude, period, phase shift, and vertical shift of the function y = sin ( x π ). Then graph the function over an interval of exactly one period. Be sure to label the top, bottom, left, and right boundaries properly. A: T: ϕ: V:

10 MA10 Trigonometry Final Exam Study Guide Page 6 15) (5 pts) Verify the identity given below, using algebraic manipulation and trigonometric identities. Be sure to show all steps of your solution. Graphical solutions are not allowed. sin x 1 + cos x cos x = csc x sin x Reasons Show the steps Reasons 16) (5 pts) Find all exact solutions for θ in degrees, 0 θ < 360, of the equation: cos θ = 1 sin θ 17) (5 pts) If a force is applied using a rope with 100 lb tension at above horizontal, find the horizontal and vertical components of the force. Round to decimal places if decimal does not terminate in fewer. Horizontal: Vertical:

11 MA10 Trigonometry Final Exam Study Guide Page 7 18) (5 pts) Verify the sum of angles formulas for sine and cosine using Euler s identity or another valid means. 19) (5 pts) Find the angle between the vectors A = 5, and B =, 5 in degrees. Round to decimal places if decimal does not terminate in fewer. 0) (5 pts) Given complex numbers M = e (38 )i and N = e (1 )i, find the following in trigonometric form: (a) MN (b) M N (c) M (d) N

12 MA10 Trigonometry Final Exam Study Guide Page 8 1) (5 pts) Translate the equation for the circle (x 1) + (y + ) = 9 into polar form. ) (5 pts) Find the area of a triangle with sides a = 10 cm, b = 1 cm, and c = 17 cm. Round to decimal places if decimal does not terminate in fewer. 3) (5 pts) Identify the equation that best matches the graph shown: (a) y = 5x + 5 (b) r = 5 sin (5θ) (c) x + y = 5 (d) (e) r = 5 5 sin θ None of these are even close!

13 MA10 Trigonometry Final Exam Study Guide Page 9 ) (5 pts) Convert (, ) from rectangular to exact polar coordinates. 5) (5 pts) Convert (, 5π/3) from polar to exact rectangular coordinates. 6) (5 pts) Find all six trigonometric functions of θ if ( 1, ) is on the terminal side of θ. sin (θ) cos (θ) tan (θ) csc (θ) sec (θ) cot (θ) 7) (5 pts) Find the linear velocity of a point on the cutting edge of a 7-inch diameter circular saw blade that is spinning at 0 revolutions per minute. Give the answer in inches per second. Round to decimal places if decimal does not terminate in fewer. 135 total points possible.

14 MA10 Trigonometry Final Exam Study Guide Page 10 Hints on How to Work These 1) Two good methods: locate angles on unit circle & use ˆθ reference angles use the sum of angles formula for tangent ) Basic identities: How is cot x related to tan x? 3) Arbitrary circle: locate the point ( 3, ) or (3, ), whichever has tan x = y x = 3 with 0 < x < π. Use the arbitrary circle definition for sine to find the exact value of sin x. ) Ambiguous case, could be 0, 1, or solutions. Use LoS to find sin β, and test it to find if there are no solutions or at least one. Apply inverse sine to find β, and test it to see if there are one or two solutions. Then use the 180 rule to find γ and LoS to find c. 5) Look up the double angle formula for sin x. What is needed to fill in the right side of the formula? Find the missing cos x using basic identity cos x + sin x = 1, being careful with ± algebraic sign from the given quadrant information. 6) Get sec x all by itself. Invert both sides of the equation to put in terms of cos x. Locate all places on the unit circle where cos x is that value. 7) Angle between the cable and the ground has opposite side = height of antenna and adjacent side = distance along ground. Use soh cah toa. 8) In right triangle, use pythagoras to find hypotenuse. (a) Use soh cah toa to get sin θ exactly, complete with radical. (b) Set your calculator to radian mode and find approximate sin 1 θ. Formula s = rθ works only with radians. 9) Look up sum of angles formula for sine. Needs sine and cosine for both angles. Use cos x + sin x = 1 to find sin y and cos x (careful about quadrants to get ± signs right). Plug in the values and calculate. 10) Non-ambiguous case of LoS, so there will be exactly one solution. Find α by the 180 rule, then a and c by the LoS. 11) Put everything in terms of sine and cosine. Multiply every term upstairs and down by sin x to clear out the fraction-within-a-fraction. Use pythagorean identity to consolidate everything in terms of cosine and simplify. 1) Formula for cosine of a half-angle requires cos x (including proper ±), and also what quadrant the half-angle is in for another ± determination. Can use a reference triangle or arbitrary circle definition to find cos x. Each quadrant for x is compressed into a halfquadrant for x. 13) Solve the oblique triangle, at least partially. LoC will find the missing side in one go.

15 MA10 Trigonometry Final Exam Study Guide Page 11 1) Formula sheet has general equation for sinusoidal function: y = A sin [ π T (x ϕ) ] + V. First put the given equation into that form by factoring out 1 so that x has no direct multiplier: y = sin ( x π ) [ ] = sin 1 (x π). A multiplier is missing, so by definition A = 1. Since it s positive, the sinusoid is not flipped. π T = 1, which can be solved for T. ϕ can be found by inspection: ϕ = π (which must be less than T or adjusted to be so) V add-on is missing, so by definition V = 0. y = V is the horizontal line that goes through the middle of the sine wave. Amplitude A is the vertical distance from the midline to the top of a hill; it is the same distance to the bottom of a valley. Period T is the horizontal distance between peaks or other identical shapes in the wave set the left bound = ϕ and then the right bound is ϕ + T. 15) General procedure: put everything in terms of sine and/or cosine simplify fraction-within-a-fraction messes combine fractions (adding with a common denominator, etc.) look for ways to simplify with cos x + sin x = 1 and factoring make the left side and right side look the same make sure everything you do is mathematically valid! 16) Need to have exactly one trig function. Here, easiest to do is put everything in terms of sine. Solve for sin θ, locate everywhere on the unit circle that sin θ equals that value or values. 17) Most people benefit from drawing a diagram to help sort out the information, especially for sorting out soh cah toa. Determine if horizontal force component is adjacent to the given angle or opposite it. Vertical force will be the other one. Assign positive values to right and up; negative values to left and down. Apply soh cah toa with the rope s force along the hypotenuse. 18) Study the steps in the answer section or use the diagram method shown in class.

16 MA10 Trigonometry Final Exam Study Guide Page 1 19) Use the formula on the formula sheet (back side, under label Vectors). The dot product upstairs can be found with a, b c, d = ac + bd. Magnitude downstairs can be found with the other formula in that same section. 0) Formulas for calculations with complex numbers are on the formula sheet under The Complex Plane (back side, right column). Use Euler s form for problems given with e to a complex power. (a) (b) (c) multiply coefficients, add angles divide coefficients, subtract angles square coefficient, double the angle (d) square-root coefficent, halve the angle to find the first square root. Then evenly space the second root at half-way around the circle (with the same coefficient). 1) Multiply out the parentheses squared. Substitute r = x + y, x = r cos θ, and y = r sin θ. ) Given three sides, use Heron s formula (on formula sheet) for the area. Remember little s is half the perimeter of the triangle. 3) Identify the graph s family by comparing with those shown in the lecture, or set your grakulator to polar mode and graph the equation. ) Find r and θ using the formulas under Polar Form on the formula sheet. Write the answer in (r, θ) form. 5) Find x and y using the formulas under Polar Form on the formula sheet. Write the answer in (x, y) form. 6) Use arbitrary circle definitions for the trig functions. 7) Find r from the diameter. Find ω by converting rpm to radians per second. Then linear velocity V = rω.

17 MA10 Trigonometry Final Exam Study Guide Page 13 1) 3 ) 3 3) 5 ) solutions: β 1 = 68.0, γ 1 = 86.60, c 1 = 35.3 cm; β = , γ = 3.0, c =.39 cm 5) 9 6) 3π, 5π 7) ) (a) 1 5 (b) cm 9) ) One solution: α = 31.00, a = 3.0 cm, c = 8.13 cm 11) (d) cot x 1) 5 13) 70 ft 1) A = 1, T = π, ϕ = π, V = π 5π 15) Show the steps sin x 1 + cos x cos x sin x 16) 90, 10, 330 = sin x + (1 + cos x) (1 + cos x) sin x = sin x cos x + cos x (1 + cos x) sin x = + cos x = (1 + cos x) sin x (1 + cos x) (1 + cos x) sin x = sin x 17) Horizontal: lb, Vertical: 0.67 lb Reasons common denominator, add fractions multiply out (1 + cos x) sin x + cos x = 1 = csc x csc x = 1 sin x factor out the and cancel (1 + cos x)

18 MA10 Trigonometry Final Exam Study Guide Page 1 18) e ix = cos x + i sin x e i(a+b) = cos (A + B) + i sin (A + B) cos (A + B) + i sin (A + B) = e i(a+b) = e ia+ib = e ia e ib cos (A + B) = cos A cos B sin A sin B sin (A + B) = sin A cos B + cos A sin B 19) = (cos A + i sin A) (cos B + i sin B) = cos A cos B + cos A i sin B + i sin A cos B + i sin A sin B = cos A cos B sin A sin B + i (sin A cos B + cos A sin B) 0) In trig form: (a) 8e (5 )i (b) e ( )i (c) 16e (76 )i (d) 1) r r cos θ + r sin θ = ) 58.9 cm² 3) You figure it out! It s not (e). ( ) 8, 3π/) ( 5), ) 3 e (7 )i, e (187 )i sin (θ) 6) 17 csc (θ) 17 7) inches/sec cos (θ) 1 17 sec (θ) 17 tan (θ) cot (θ) 1

19 Note: This formula page may contain some formulas that are not needed for this exam. Quadratic Formula x = b ± b ac a Arcs and Sectors s = rθ A = 1 θr Linear and Angular Velocity V = s t ω = θ t V = rω s = rθ General Equation of a Sinusoidal Function y = A sin [ π T (x ϕ)] + V Sum and Dierence of Angles Formulas sin (α ± β) = sin α cos β ± cos α sin β cos (α ± β) = cos α cos β sin α sin β tan (α ± β) = tan α ± tan β 1 tan α tan β Double-Angle Formulas sin α = sin α cos α cos α = cos α sin α = cos α 1 = 1 sin α tan α = tan α 1 tan α = cot α cot α 1 = cot α tan α Half-Angle Formulas sin α = ± 1 cos α cos α = ± 1 + cos α tan α = 1 cos α sin α = sin α 1 + cos α Product-to-Sum Formulas sin α cos β = 1 [sin (α + β) + sin (α β)] cos α sin β = 1 [sin (α + β) sin (α β)] sin α sin β = 1 [cos (α β) cos (α + β)] cos α cos β = 1 [cos (α + β) + cos (α β)] Sum-to-Product Formulas sin α + sin β = sin α + β sin α sin β = cos α + β cos α + cos β = cos α + β cos α cos β = sin α + β Law of Sines a sin α = b sin β = Law of Cosines c sin γ a = b + c bc cos α b = a + c ac cos β c = a + b ab cos γ Area of Triangles S = 1 bc sin α S = 1 ac sin β S = 1 ab sin γ S = a sin β sin γ sin α S = b sin α sin γ sin β S = c sin α sin β sin γ or sin α a cos α β sin α β cos α β sin α β = sin β b = sin γ c S = s (s a) (s b) (s c), s = a + b + c

20 Mollweide's Equation (a b) cos γ = c sin α β Vectors Magnitude: a, b = a + b Angle between: cos θ = Polar Form u v u v x + y = r or r = x + y x = r cos θ or θ = cos 1 x r y = r sin θ or θ = sin 1 y r tan θ = y x or θ = tan 1 y x. The Complex Plane cisx = cos x + i sin x a + bi = rcisθ = re iθ Given complex numbers A = acisα and B = bcisβ: AB = abcis (α + β) A B = a cis (α β) b A n = a n cis (nα) Euler's form: A = ae iα and B = bc iβ AB = abe (α+β)i A B = a b e(α β)i A n = a n e nα