الأسبوع 14 · التمثيل البياني لدالتي الجيب وجيب التمام، وإيجاد قياس زاوية بمعلومية إحدى نسبها المثلثية · 3 دروس
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The graph of the cosine function and of y = a sin θ and y = a cos θ: amplitude, range, and a depth-of-water model
Mathematics Sec 1 Term 1, Unit 4, Lessons 5–6
GoalDraw y = cos θ, find the amplitude of y = a sin θ and use it in a water model.
New words
amplitude · half the distance between the highest and the lowest value, |a| for y = a sin θperiodic function · a function whose graph repeats after a fixed period
Explanation
The graph of y = cos θ is the same wave as the sine, but it starts at its maximum: cos 0 = 1, then 0 at π/2, −1 at π, 0 at 3π/2 and 1 at 2π. Its domain is all real numbers, its range [−1, 1] and its period 2π. The graph is symmetrical about the y-axis because cos(−θ) = cos θ. In y = a sin θ and y = a cos θ the number a stretches the wave up and down: the amplitude is |a|, the range is [−|a|, |a|], and the period stays 2π. If a is negative the wave is also turned upside down. Wave model: the surface of the water in a wave tank rises and falls so that its height above the calm level is h = 0.4 sin θ metres. The amplitude is 0.4 m, so with a calm depth of 2 m the depth varies between 2 − 0.4 = 1.6 m and 2 + 0.4 = 2.4 m.
Worked examples
State the amplitude and range of y = 3 cos θ: amplitude 3, range [−3, 3], period 2π.
y = −2 sin θ has amplitude |−2| = 2 and range [−2, 2]. Compared with y = 2 sin θ it is turned upside down: it goes down first.
ƒFormula
h = 0.4 sin θ
the water surface height above its calm level, in metres; the depth is 2 + h
Wave tank: h = 0.4 sin θ m with θ = (π/3)t and t in seconds. The period is 2π ÷ (π/3) = 6 s. At t = 1.5 s, θ = π/2 and h = 0.4 m, so the depth is 2.4 m, the maximum. At t = 4.5 s, θ = 3π/2 and the depth is 1.6 m, the minimum.
Class activity
Slinky wave. Tie a long rope to a post, hold the other end and shake it up and down to make waves. Measure how high the crest rises above the rest line (the amplitude) and how far apart two crests are. Shake harder and see what changes.
Practice
1
Give the range of y = 5 sin θ and of y = −0.5 cos θ.
[−5, 5] and [−0.5, 0.5].
2
Find cos(3π) and cos(−π/2).
cos 3π = cos π = −1; cos(−π/2) = 0.
3
In the tank model, find the depth when θ = π/6.
h = 0.4 × ½ = 0.2 m, depth 2.2 m.
4
A swing’s horizontal displacement is x = 0.8 cos θ metres. What are the largest displacements to each side?
0.8 m to the right and 0.8 m to the left (amplitude 0.8).
5
How many times does y = cos θ equal 1 for 0 ≤ θ ≤ 4π?
