5.5 Make Connections With Sine Functions 

Prerequisite Skills 

1. Sketch a graph of each function. Then, write the domain and range for each function.

a) y = -x2 - 5 b) y = 2(x + 1)2

1. The depth of water, d(t) metres, in a seaport can be approximated by the equation d(t) = 1.5 sin(29.52t - 44.28) + 12.4, where t is the time in hours (t = 0 at 12:00 A.M.).

a) Use a graphing calculator to graph the function for 24 h.

b) Determine high tide and when it occurs.

c) Determine low tide and when it occurs.

d) Determine the period of the function.

e) A cruise ship needs at least 12 m of water to dock safely. For how many hours in each period can the ship dock safely? Round your answer to the nearest tenth of an hour.

2. An object attached to the end of a spring is oscillating up and down. The displacement of the object is given by y = 2.4 sin(12t + 90.6), where t is the time, in seconds, and y is the distance, in centimetres. When the displacement is 0. the spring is at its relaxed length.

a) Sketch a graph of the function for 60 s.

b) What is the displacement from the spring's relaxed length after 12 s?

c) What is the amplitude of the function? In what units is the amplitude measured?

3. A hand crank for a window rotates such that the height of the handle, h, in centimetres is given by h(θ) = 4 sin(θ + 45°), where θ is the rotational angle relative to the horizontal.

a) Sketch a graph of this function for two rotations.

b) At what height was the handle when the rotation began?

c) What was the position of the handle relative the horizontal, when the rotation began?

4. A bridge will sway slightly if a large number of automobiles drive over it at once. During a period of heavy traffic, the displacement of the bridge's centre, in centimetres, relative to the normal position, is modelled by f(t) = 2.1 sin(47t), where t is the time, in seconds.

a) Sketch a graph that shows the swaying of the bridge over 30 s.

b) What is the displacement after 2 s?

c) How wide is the sway? d) What is the period of the sway?

5. Without the use of technology, sketch a graph of each function for one period. 

a) f(x) = sin(x – 90°) - 3 

b) f(x) = - ¼ sin x + 1