Trigonometric functions: differentiation, investigation and models · Grade 11 · 5 units
In Trigonometric functions: differentiation, investigation and models (Grade 11 · 5 units) you practise 7 skills: The parameters of a wave · The derivatives of sine, cosine and tangent · Investigating a trigonometric function · Tangents to the graphs of sine and cosine · Integrals and areas · Periodic models · Periodic functions. Sine and cosine repeat after every whole turn, so they suit things that come back again and again: a big wheel at a fair, the tides, a population that grows and shrinks with the seasons. In this topic angles are measured in radians, and a whole turn is two pi.
In a function made of a number times a sine, plus a number, the number in front stretches the wave up and down, and the number added moves it up or down. The number that multiplies the variable inside the sine sets how many waves fit in one turn: the period is two pi divided by that number. The derivative of sine is cosine, and the derivative of cosine is minus sine. With them you find the gradient of a tangent, the turning points and the largest and smallest values, just as for other functions. The integrals follow too: an antiderivative of cosine is sine, and of sine it is minus cosine.
Tip: When looking for the largest value on an interval, compare the values where the derivative is zero with the values at the ends of the interval.
What you learn
The parameters of a wave
For example: What is the period of the function f(x) = sin(x / 2) ?
Answer and solution
The answer: 4π
The function repeats when x / 2 goes through a whole turn, 2π . So the period is 2π × 2 = 4π .
The derivatives of sine, cosine and tangent
For example: The function is f(x) = sin x . What is f′(5π/3) ?
Answer and solution
The answer: 1/2
The derivative is f′(x) = cos x , so f′(5π/3) = cos(5π/3) , that is 1/2 .
Investigating a trigonometric function
For example: At which x in 0 ≤ x ≤ 2π does the function f(x) = sin x + cos x take its largest value?
Answer and solution
The answer: π/4
The derivative is f′(x) = cos x − sin x . It is 0 when sin x = cos x , that is at x = π/4 and at x = 5π/4 . There f(π/4) = √2 and f(5π/4) = −√2 , and at the ends of the interval the value is 1. So the largest value is taken at x = π/4 .
Tangents to the graphs of sine and cosine
For example: What is the gradient of the tangent to the graph of f(x) = sin x at the point where x = π ?
Answer and solution
The answer: −1
The gradient is f′(π) = cos(π) , that is −1 .
Integrals and areas
For example: What is the value of ∫₀^(π/2) sin x dx ?
Answer and solution
The answer: 1
An antiderivative of sin x is −cos x . At the upper bound it is 0 , and at the lower bound −1 . So 0 − (−1) = 1 .
Periodic models
For example: A big wheel with diameter 40 metres turns once a minute, and its centre is 30 metres above the ground. The height of a cabin after t minutes is h(t) = 30 + 20 sin(2πt) metres. What is the greatest height of the cabin?
Answer and solution
The answer: 50
The greatest height comes when the sine is 1 : 30 + 20 = 50 metres.
Periodic functions
For example: What is the period of the function f(x) = sin 2x + cos 2x ?
Answer and solution
The answer: π
Both terms have the same period, 2π ÷ 2 , so the sum has it too: π .
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).