C. HECKMAN TEST 2A SOLUTIONS 170

Transcription

1 C HECKMN TEST SOLUTIONS 170 (1) [15 points] The angle θ is in Quadrant IV and tan θ = Find the exact values of 5 sin θ, cos θ, tan θ, cot θ, sec θ, and csc θ Solution: point that the terminal side of the angle θ passes through is (5, ), so we have x = 5 and y = Thus, r = x + y = 9 Note that, since r 1, the point (x, y) is not on the unit circle, so we need to use the definitions from section 44: sin θ = y r = 9 cos θ = x r = 5 9 tan θ = y x = 5 sec θ = r x = 9 5 csc θ = r y = 9 cot θ = x y = 5 Grading: + points for finding r, + points for each trig function Grading for common mistakes: points for a sign error, points for including decimal approximations () [15 points] In a right triangle BC (with the right angle at C), α = 7 and c = 16 Find a, b, and β; round answers to four decimal places Solution: First, the easy one: β = 90 α = 5 Now we use the SOHCHTO mnemonic to find a and b: sin α = a c, so a = 16 sin(7 ) = 9690, cos α = b c, so b = 16 cos(7 ) = 1778 Grading: +5 points for each answer Grading for common mistakes: 5 points if c was in the wrong place; points if a and b were backwards; points if the calculation was done in radians 1

2 () [15 points] The sinusoidal function f(x) = sin(bx ( + C) ) has period π, phase shift π 5π, and the graph of f(x) passes through the point 4, 4 Find a formula for f(x) Solution: The period of the sinusoidal function is π B, so π π = π and B = B π = The phase shift is C B, so C / = π and C = π = π To find, we substitute x = 5π 4 f into the general formula: ( ) 5π 4 = sin (B 5π4 ) + C ) ( 4 = sin 5π 4 π ( π ) = sin =, ( so = 4 and f(x) = 4 sin x π ) Grading: = 5 points for finding each of, B, C mistakes: 1 point for saying that was the amplitude Grading for common (4) [15 points] n airplane is flying horizontally 5,000 feet above the surface of the earth n observer on the surface notices that the airplane is at an angle of 50 from the zenith ( directly above ) How far away from the observer is the plane? Solution: We sketch a picture of the situation (shown below) and look for right triangles; then we use SOHCHTO to find any missing values plane 5,000 x 50 observer From the above picture, it is clear that cos(50 ) = 5,000, so x = 5,000 x cos(50 ) = 8,8910 feet Grading: +5 points for the picture, +5 points for cos(50 ), and +5 points for finding x Grading for common mistakes: points for making the angle between the plane and the horizon 50 instead; 5 points for finding the horizontal distance; points if the calculator was in radian mode

3 (5) [15 points] On a carousel (merry-go-round), the outer row of animals is 5 feet from the center Sally gets on one of the animals, and the ride starts up The ride has mechanical difficulties and has to stop after it has rotated 600 How far did Sally travel on the carousel? Solution: The path that Sally travels is a circular arc, so we use the formula s = rθ Here, r = 5, and we must convert 600 to radians to get θ: θ = π 600, so 180 Sally travels a distance of s = rθ = 5 π 50π 600 = 180 or 6180 feet Grading: +5 points for s = rθ, +5 points for converting to radians, +5 points for substitution ( ( )) 4 (6) [15 points] Find cos sin 1, where > 0 Your answer should not include any trigonometric functions or inverse trigonometric functions ( ) 4 Solution: If we let θ = sin 1, then we know that sin θ = 4, and that θ is in Quadrant I or Quadrant IV Since we are told that > 0, we must have θ in Quadrant I, so we can draw a right triangle, whose opposite side is 4 and whose hypotenuse is : 4 x θ To find cos θ, we need the value of x We find that by using the Pythagorean Theorem: x + 4 =, so x = 16 (x must be positive, to make cos θ positive) Then cos θ = x = 16 ), +5 points for drawing a ( 4 Grading: +5 points for deciphering sin 1 triangle (or doing something equivalent), +5 points for finding cos θ

4 (7) [10 points] Verify the following trigonometric identity [You do not need to enter this answer on the Summary of nswers page] cot t cos t = csc t Solution: Let s rewrite cot t and csc t in terms of and cos t, then do some algebra on the left-hand side: cot t + cos t cos t 1 + cos t cos t(1 + cos t) + () (1 + cos t) cos t + cos t + sin t (1 + cos t) cos t + 1 (1 + cos t) 1 csc t 1 Variations are also possible Grading: Done on a point basis Generally, points were awarded if substitutions were made, or algebra was done; 5 points were awarded if both were done, but the student appeared lost as what to do; 7 points if some indication of purpose was shown 4

5 C HECKMN TEST B SOLUTIONS 170 (1) [15 points] The angle θ is in Quadrant III and cos θ = Find the exact values of 7 sin θ, cos θ, tan θ, cot θ, sec θ, and csc θ Solution: point that the terminal side of the angle θ passes through is (x, y), where x = and r = 7 Since θ is in Quadrant III, y = r x = 40 sin θ = y r = 40 7 cos θ = x r = 7 tan θ = y x = 40 sec θ = r x = 7 csc θ = r y = 7 40 cot θ = x y = 40 Grading: + points for finding y, + points for each trig function Grading for common mistakes: points for a sign error, points for including decimal approximations () [15 points] In a right triangle BC (with the right angle at C), α = 5 and c = 5 Find a, b, and β; round answers to four decimal places Solution: First, the easy one: β = 90 α = 65 Now we use the SOHCHTO mnemonic to find a and b: sin α = a c, so a = 5 sin(5 ) = , cos α = b c, so b = 5 cos(5 ) = 6577 Grading: +5 points for each answer Grading for common mistakes: 5 points if c was in the wrong place; points if a and b were backwards; points if the calculation was done in radians 1

6 () [15 points] The sinusoidal function f(x) = sin(bx + C) has period π, phase shift π (, π ) and the graph of f(x) passes through the point 1, Find a formula for f(x) Solution: The period of the sinusoidal function is π B, so π B = π and B = π π = The phase shift is C B, so C / = π and C = π = π substitute x = π into the general formula: 1 ( π ( f = sin B 1) π ) 1 + C ( = sin π 1 π ) ( so = and f(x) = sin x π ) ( = sin π ) =, To find, we Grading: = 5 points for finding each of, B, C mistakes: 1 point for saying that was the amplitude Grading for common (4) [15 points] n airplane is flying horizontally 5,000 feet above the surface of the earth n observer on the surface notices that the airplane is at an angle of 55 from the zenith ( directly above ) How far away from the observer is the plane? Solution: We sketch a picture of the situation (shown below) and look for right triangles; then we use SOHCHTO to find any missing values plane 5,000 x 55 observer From the above picture, it is clear that cos(55 ) = 5,000, so x = 5,000 x cos(55 ) = 61,0064 feet Grading: +5 points for the picture, +5 points for cos(55 ), and +5 points for finding x Grading for common mistakes: points for making the angle between the plane and the horizon 55 instead; 5 points for finding the horizontal distance; points if the calculator was in radian mode

7 (5) [15 points] On a carousel (merry-go-round), the outer row of animals is 0 feet from the center Sally gets on one of the animals, and the ride starts up The ride has mechanical difficulties and has to stop after it has rotated 840 How far did Sally travel on the carousel? Solution: The path that Sally travels is a circular arc, so we use the formula s = rθ Here, r = 0, and we must convert 840 to radians to get θ: θ = π 840, so 180 Sally travels a distance of s = rθ = 0 π 80π 480 = 180 or 9 feet Grading: +5 points for s = rθ, +5 points for converting to radians, +5 points for substitution ( ( )) (6) [15 points] Find an 1, where > 0 Your answer should not include any trigonometric functions or inverse trigonometric functions ( ) Solution: If we let θ = tan 1, then we know that tan θ =, and that θ is in Quadrant I or Quadrant IV Since we are told that > 0, we must have θ in Quadrant I, so we can draw a right triangle, whose opposite side is and whose adjacent side is : x θ To find sin θ, we need the value of x We find that by using the Pythagorean Theorem: + = x, so x = + 4 (The hypotenuse is always positive) Then sin θ = x = + 4 ( ) Grading: +5 points for deciphering tan 1, +5 points for drawing a triangle (or doing something equivalent), +5 points for finding sin θ

8 (7) [10 points] Verify the following trigonometric identity [You do not need to enter this answer on the Summary of nswers page] (tan x + 1)(cos x + 1) = tan x + Solution: Let s FOIL out the left-hand side, and use trigonometric identities to make it look like the right-hand side: (tan x + 1)(cos x + 1) tan x + tan x cos x + tan x + cos x + 1 sin x cos x cos x + tan x + cos x + 1 sin x + cos x + tan x tan x + 1 Variations are also possible Grading: Done on a point basis Generally, points were awarded if substitutions were made, or algebra was done; 5 points were awarded if both were done, but the student appeared lost as what to do; 7 points if some indication of purpose was shown 4