2. Determine the amplitude of:
y= cos x
3. Determine the domain of:
y= 4 sin x
4. Determine the range of:
y= - sin x
5. Determine the period of:
y= 3 sin ðx
6. Determine the phase shift of:
y= 3 cos (2x -1)
7. Solve the following equation for indicated value of x:
y= 4 sin x; for x = ð
8. Solve the following equation for the indicated value of x
y= - cos x ; for x= 4ð
Most of these questions can be answered by the definition of the general trigonometric functions:
y = a sin A(x-φ)
a is the amplitude
2π/A = period
φ = phase shift.
If you can transform the trigonometric functions into the above form and evaluate the values of a, A, and φ, the answers would become apparent.
2. The amplitude of a trigonometric function can be determined by looking at the coefficient in front of the trigonometric function. In the case of y = cos(x), there is no coefficient, so the amplitude is 1.
3. The domain of a trigonometric function includes all real numbers, unless there are restrictions due to the nature of the function. In the case of y = 4sin(x), the domain is all real numbers because there are no restrictions on the angle x.
4. The range of a trigonometric function can be determined by looking at the range of the basic trigonometric function it is based on. In the case of y = -sin(x), the basic sine function has a range between -1 and 1. Multiplying by -1 in this case reverses the range, so the range of y = -sin(x) is between -1 and 1.
5. The period of a trigonometric function is the distance between two consecutive peaks or troughs of the function. For the basic sine function y = sin(x), the period is 2π. In the case of y = 3sin(θx), where θ is a constant, the period is 2π/θ. So, in the case of y = 3sin(θx), the period is 2π/3.
6. The phase shift of a trigonometric function determines the horizontal shift of the graph. In the case of y = 3cos(2x - 1), the phase shift is determined by the constant in the parentheses. The formula for phase shift is -(constant in parentheses)/coefficient of x. Therefore, the phase shift in this case is -(-1)/2 = 1/2 to the right.
7. To solve the equation y = 4sin(x) for x = π, substitute π into the equation and solve for y. y = 4sin(π) = 4(0) = 0. Therefore, when x = π, y = 0.
8. To solve the equation y = -cos(x) for x = 4π, substitute 4π into the equation and solve for y. y = -cos(4π) = -1. Therefore, when x = 4π, y = -1.