# Trigonometric functions: differentiation, investigation and models · Grade 11 · 5 units

> Explanation, worked examples and free practice following the Israeli curriculum.

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

- Example: What is the period of the function f(x) = sin(x / 2) ?
- Answer: 4π
- Solution: 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

- Example: The function is f(x) = sin x . What is f′(5π/3) ?
- Answer: 1/2
- Solution: The derivative is f′(x) = cos x , so f′(5π/3) = cos(5π/3) , that is 1/2 .

### Investigating a trigonometric function

- Example: At which x in 0 ≤ x ≤ 2π does the function f(x) = sin x + cos x take its largest value?
- Answer: π/4
- Solution: 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

- Example: What is the gradient of the tangent to the graph of f(x) = sin x at the point where x = π ?
- Answer: −1
- Solution: The gradient is f′(π) = cos(π) , that is −1 .

### Integrals and areas

- Example: What is the value of ∫₀^(π/2) sin x dx ?
- Answer: 1
- Solution: 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

- 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: 50
- Solution: The greatest height comes when the sine is 1 : 30 + 20 = 50 metres.

### Periodic functions

- Example: What is the period of the function f(x) = sin 2x + cos 2x ?
- Answer: π
- Solution: 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. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 11 · 5 units topics](https://kixplo.com/il/en/math/grade-11-5u/)