MATH EXAM 1 - SPRING 2018 SOLUTION

Transcription

1 MATH EXAM 1 - SPRING 018 SOLUTION 8 February 018 Instructor: Tom Cuchta Instructions: Show all work, clearly and in order, if you want to get full credit. If you claim something is true you must show work backing up your claim. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct. Justify your answers algebraically whenever possible to ensure full credit. Circle or otherwise indicate your final answers. Please keep your written answers brief; be clear and to the point. Good luck!

2 1. ( points Sketch a graph of the equation y = (x +. Include at last two labeled points in your plot to guarantee full credit. y = x y = (x h.shift 10 (1, 1 4 (0, 0 4 (, 0 ( 1, 1 4 y = (x + 1 v.shift 10 4 ( 1, (,. ( points Draw the specified angle. (a ( points 4π radians (b ( points 189 (c ( points π radians

3 . ( points Here is a portion of the unit circle, partially labeled: y (, θ (1, 0 x (a ( points What is sin(θ? sin(θ = (b ( points What is tan(θ? tan(θ = (c ( points What is sec(θ? sec(θ = 4. (1 points (a (4 points Convert 4 to radians. ( π radians 4 = (4 0 = (4π 0 (b (4 points Convert π radians to degrees. 11 π 11 radians = ( ( π 0 11 radians = π radians radians = radians. ( (0 = (11 (c (4 points Convert 4 radians to degrees. ( ( radians = (4 radians =. π radians π. (1 points Find an exact value for... (a (4 points cos Since the coordinates of the angle π on the unit circle are (, 1 cos =. ( , we see that

4 (b (4 points cot Since the coordinates of the angle π on the unit circle are (0, 1, we see that cot = 0 1 = 0. ( 11π (c (4 points sin Since the coordinates of 11π on the unit circle are ( 11π sin = 1. (, 1, we see that. ( points Solve the triangle. A b 1 Find A We use the fact that 1 + }{{} 90 right angle +A = 180. Solving for A yields Find b a A = = 90 1 = 9. We use the sine function to write sin(1 = b. Therefore solving for b yields b = sin(1. Find a We use the cosine function to write cos(1 = a. Therefore solving for a yields a = cos(1.. (8 points (a (4 points Solve the equation x + 4 = 9. Subtract 4 to get x = 9 4 =. Divide by to get x =. (b (4 points Solve the equation x + x = 0. Factor the left-hand side to get (x + (x = 0. Using the zero product property of the real numbers, we may conclude that either x + = 0 or x = 0. Thus either x = or x =.

5 8. (1 points If cos(t = and t is in quadrant IV, find the other five trigonometric functions. 9 Since we are told that cos(t =, we draw a triangle in which that occurs: 9 9 A? t To find the other trigonometric functions of t, we must find the side labelled?: Find? Using the Pythagorean theorem, we write +? = 9, so we get 4+? = 81. Subtract 4 and get? = and take the square root to get? = ±. We must throw away the negative solution because it is physically meaningless (negative side length in a triangle doesn t make sense. Therefore? =. Also we are told that t is in quadrant IV. This means that sine, tangent, cosecant, and cotangent will be negative while cosine and second will be positive. Now we may answer the question. The other five trig functions sin(t =? 9 = 9, tan(t =? =, sec(t = 9, csc(t = 9? = 9, and cot(t =? =.

6 9. (8 points Find the radius a circle must have if an arc length of is subtended by an angle of θ = 1. Express your final answer accurate to at least two decimal places. We want to use the formula s = rθ, but to do so, θ must be expressed in radians. So first we must convert θ to radians: ( π radians 1 = (1 0 = π 1 radians. Therefore we can now use the formula s = rθ with s = and θ = π, and then we can use it to solve 1 for r (as requested. Plugging in: = (r. 1 To solve for r, divide both sides by π 1 r = π 1 to get = 1 ( 1 = 10 π π (1 points There is an antenna on the top of a building. From a location feet from the base of the building, the angle of elevation to the top of the building is measured to be 10. From the same location, the angle of elevation to the top of the antenna is measured to be 1. Find the height of the antenna. Express your final answer accurate to at least two decimal places. We start with a figure to describe the problem: H A H B 1 10 Thus, we seek H A. Note that we have two right triangles in this picture, reproduced (in a simplified form below: H B H B + H A 10 1 Thus, tan (10 = H B H B = tan (10 and tan (1 = H A + H B H A + H B = tan (1

7 So we see H A = (H A + H B H B = tan (1 tan (10 a 11.9 feet. 11. (4 points Use your calculator to compute the following trig functions accurate to at least two decimal places. (a (1 point sin(1 ( 9π (b (1 point cos 1 (c (1 point sec(11 (d (1 point sin(1 sin( cos sec(11 = ( π 1 cos( sin(1 0.8.