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Graphs of Sine and Cosine
Trigonometric Functions and Graphs
Conceptual 1
Conceptual 2
Conceptual 3
100
Identify the amplitude and period of the function g(x) = (1/3) sin x.
Amplitude = 1/3; Period = 2∏
100
Find the frequency in hertz for the sound wave with a period of 0.01 second and amplitude of 6 in.
What is 100
100
The graph of y = -A cos(-Bx) is the graph of y = Acos(Bx)
What is False. Observe that since cosine is even, y = −Acos(−Bx) = −Acos(Bx) , which is a reflection of y = Acos(Bx) over the x-axis.
100
TRUE or FALSE? sin(2n∏ + Ø) = sin Ø, n an integer
What is TRUE The angles (2n∏ + Ø) and Ø are coterminal for any integer n. Hence, they have the same cosine and sine values.
100
TRUE or FALSE: tan (Ø + 2n∏) = tan Ø, n an integer
What is TRUE because tan(Ø + 2n∏) = tan[ (Ø + n∏) + n∏] = tan(Ø + n∏) = tanØ
200
The population of field mice can be modeled by y = 3000 + 1340 sin (π/6) * t where y represents the number of mice and t represents the number of months past March 1 of a given year. What is the maximum number of mice?
4340
200
Find the frequency in hertz for the sound wave with a period of 0.025 second and amplitude of 5 in.
What is 40
200
The graph of y = -A sin(-Bx) is the graph of y = A sin(Bx)
What is True. Observe that since sine is odd, y = −Asin(−Bx) = −(−Asin(Bx)) = Asin(Bx) .
200
TRUE or FALSE? cos(2n∏ + Ø) = cos Ø, n an integer
What is TRUE
200
Is cosecant an even or an odd function? Justify your answer.
What is odd function
300
Suppose the tidal range of a city on the Atlantic coast is 18 feet. A tide is at equilibrium when it is at its normal level, halfway between its highest and lowest points. Each tide cycle lasts about 12 hours. Write a function to represent the height h of the tide. Assume that the tide is at equilibrium at t = 0 and that the high tide is beginning.
h = 9 sin (π/6)t
300
Identify the x-intercepts and phase shift of the function g(x) = cos (x – 2∏).
What is x-intercepts for cos (x – 2∏) will occur at 2∏ + ∏/2 + n∏ = 5∏/2 + n∏.
300
Find the y-intercept of the function y = A cos (Bx). A and B are positive real numbers.
Since Acos (B×0) = A(1) = A , the y-intercept of y = Acos(Bx) is (0, A).
300
sinØ = 1 when Ø = [ (2n + 1)*∏ ] / 2, n an integer
What is FALSE. False. For instance, sin (3∏/2) = -1 (which corresponds to n = 1).
300
Is tangent an even or an off function? Justify your answer.
What is an odd function.
400
Doctors may use a tuning fork that resonates at a given frequency as an aid to diagnose hearing problems. The sound wave produced by a tuning fork can be modeled using a sine function. If the amplitude of the sine function is 0.25, write the equation for tuning forks that resonate with a frequency of 512 Hertz.
f = 0.25 sin (1024π)*t
400
Identify the equation for a sine function of period 90°, after a phase shift 20° to the left.
What is y = sin[4(x + 20°)]
400
Find the y-intercept of the function y = A sin (Bx)
Since Asin (B×0) = A(0) = 0 , the y-intercept of y = Asin(Bx) is (0, 0).
400
cos Ø = 1 when Ø = n∏, n an integer
What is FALSE. For instance, when cos ∏ = -1 (which corresponds to n = 1).
400
TRUE or FALSE? Given A and B are positive real numbers. The graph of y = - A cosBx is the graph of y = AcosBx reflected about the x-axis.
What is True. In general, the graphs of y = f (x) and y = − f (x) are reflections of each other over the x-axis.
500
The population of field mice can be modeled by y = 3000 + 1340 sin π/6 · t where y represents the number of mice and t represents the number of months past March 1 of a given year. Determine the period of the function.
What is 12
500
The Ferris wheel at the landmark Navy Pier in Chicago takes 7 minutes to make one full rotation. The height H in feet above the ground of one of the six-person gondolas can be modeled by H(t) = 45 sin (2∏/7)(t – 1.75) + 40, where t is time in minutes. Find the height of a cabin to find what is the maximum height of a cabin?
What is 85 feet
500
What is the range of y = 2A sin(Bx + C) - A/2 ?
First, note that the amplitude of y = 2Asin(Bx +C) . The range of this function is [-2A, 2A]. Since subtracting 2 A from it shifts the graph down that many units, we do the same to the endpoints of the range of the non-translated function to obtain [-5A/2 , 3A/2]
500
TRUE or FALSE tan Ø = 0 if and only if Ø = = [ (2n + 1)*∏ ] / 2, n an integer.
What is FALSE because tan Ø = 0, which means that sin Ø = 0 => Ø = n∏
500
The graph of y = A sin(-Bx) is the graph of y = A sin(Bx) reflected about the x-axis.
What is True. Observe that since sine is odd, y = Asin(−Bx) = −Asin(Bx) , which is a reflection of y = Asin(Bx) over the x-axis.