2. A displacement is written as x(t) = 8 cos(omega t + 5pi/6). Which sine form represents the same function for every t?
A.
8 sin(omega t + pi/3)
B.
8 sin(omega t - pi/3)
C.
8 sin(omega t + 4pi/3)
D.
8 sin(omega t - 5pi/6)
Correct Answer:
8 sin(omega t + pi/3)
Explanation:
Since cos(theta) = sin(theta + pi/2), the phase becomes 5pi/6 + pi/2 = 4pi/3, which is equivalent modulo 2pi to -2pi/3, not listed. Rechecking with sin(theta) = cos(theta - pi/2), cos(omega t + 5pi/6) = sin(omega t + 5pi/6 + pi/2) = sin(omega t + 4pi/3). Therefore the correct option is 8 sin(omega t + 4pi/3).
3. Which value of phi makes the identity 3 sin(2t - pi/4) = 3 cos(2t + phi) valid for all t?
A.
-3pi/4
B.
pi/4
C.
3pi/4
D.
5pi/4
Correct Answer:
-3pi/4
Explanation:
Because sin(theta) = cos(theta - pi/2), the cosine phase must be -pi/4 - pi/2 = -3pi/4. Any phase differing by an integer multiple of 2pi is equivalent.
4. The function x(t) = -6 sin(3t + pi/5) is to be expressed with a positive amplitude as a cosine. Which form is equivalent?
5. Two representations of a periodic function are x(t) = A sin(omega t + alpha) and x(t) = A cos(omega t + beta), where A is positive. Which relation between alpha and beta is necessary and sufficient?
A.
beta = alpha + pi/2 modulo 2pi
B.
beta = alpha - pi/2 modulo 2pi
C.
beta = -alpha modulo 2pi
D.
beta = alpha + pi modulo 2pi
Correct Answer:
beta = alpha - pi/2 modulo 2pi
Explanation:
Using sin(theta) = cos(theta - pi/2), A sin(omega t + alpha) = A cos(omega t + alpha - pi/2). Hence beta = alpha - pi/2 modulo 2pi.
6. Which expression is identically equal to 7 cos(5t - 2pi/3)?
A.
7 sin(5t - pi/6)
B.
7 sin(5t + pi/6)
C.
7 sin(5t - 7pi/6)
D.
7 sin(5t + 5pi/6)
Correct Answer:
7 sin(5t - pi/6)
Explanation:
Using cos(theta) = sin(theta + pi/2), the sine phase is -2pi/3 + pi/2 = -pi/6. Therefore the equivalent expression is 7 sin(5t - pi/6).
7. For the identity 4 cos(6t - pi/7) = 4 sin(6t + phi), which value of phi in the interval 0 <= phi < 2pi is correct?
A.
3pi/14
B.
5pi/14
C.
11pi/14
D.
17pi/14
Correct Answer:
5pi/14
Explanation:
Since cos(theta) = sin(theta + pi/2), phi = -pi/7 + pi/2 = 5pi/14. This lies in the specified interval.
8. A function has the two forms x(t) = 10 sin(omega t - 4pi/3) and x(t) = 10 cos(omega t + phi). Which value of phi gives an equivalent form in the interval -pi < phi <= pi?
A.
-5pi/6
B.
-pi/6
C.
pi/6
D.
5pi/6
Correct Answer:
-5pi/6
Explanation:
Converting sine to cosine gives phi = -4pi/3 - pi/2 = -11pi/6. Adding 2pi yields phi = pi/6, which lies in the interval. Therefore the correct option is pi/6.
9. Which phase shift makes the two functions 2 sin(9t + pi/8) and 2 cos(9t + phi) identical for all t?
10. The function x(t) = -9 cos(2t + pi/6) is required in the form 9 sin(2t + phi), with positive amplitude. Which phase is valid modulo 2pi?
A.
phi = -pi/3
B.
phi = pi/3
C.
phi = 2pi/3
D.
phi = 5pi/3
Correct Answer:
phi = 2pi/3
Explanation:
Since -cos(theta) = sin(theta - pi/2), the phase is pi/6 - pi/2 = -pi/3, which is equivalent to 5pi/3 modulo 2pi. Therefore the correct option is phi = 5pi/3.
Short Questions and Answers
Practice short questions and answers related to
Sine-Cosine Equivalence
.
Short questions are currently unavailable for this page.