# Trigonometric functions · Grade 10 · 5 units

> Trigonometric functions, Grade 10 · 5 units: A number on the unit circle and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Trigonometric functions (Grade 10 · 5 units) you practise 9 skills: A number on the unit circle · Sine and cosine as coordinates · Identities from the unit circle · Special values · The graphs of sine and cosine · The tangent · When there is a solution · Trigonometric equations · A periodic model.

On the unit circle, centred at the origin with a radius of 1, every real number has a point. Start at the point where x is 1 and go along the circle a distance equal to the number: anticlockwise when the number is positive, clockwise when it is negative. The circumference is two pi, so a whole turn is two pi, half a turn is pi and a quarter turn is pi over two. Numbers that differ by whole turns reach the same point.

The cosine of a number is the x coordinate of its point, and the sine is its y coordinate. So both are always between minus one and one, and both repeat after two pi. The tangent is the sine over the cosine, and it is not defined where the cosine is 0. The circle also shows identities: the sine of the opposite number is minus the sine, and for every number the sine squared plus the cosine squared is 1, by Pythagoras’ theorem.

To solve an equation such as sine x equals a half, find one number that is a solution, pi over six, then the other number on the circle with the same y coordinate, five pi over six. The general solution adds an integer multiple of two pi to each of them: the multiple can be positive, negative or 0. A big wheel, the tides and the length of the day over a year all repeat, so they are described with a sine.

A model proof of the identity: the point of a number on the unit circle has the cosine as its x coordinate and the sine as its y coordinate. Drop a perpendicular from the point to the horizontal axis. It makes a right triangle whose legs are the size of the cosine and the size of the sine, and whose hypotenuse is the radius, 1. By Pythagoras’ theorem the sine squared plus the cosine squared is 1. A square does not depend on the sign, so the identity holds in every quadrant, and on the axes you can check it directly.

**Tip:** Sketch the unit circle and mark the point on it: the signs of the sine and the cosine show at once from the quadrant.

## What you learn

### A number on the unit circle

- Example: Which point on the unit circle belongs to the number π/2 ? (The unit circle)
- Answer: (0, 1)
- Solution: The number π/2 is a quarter turn anticlockwise, so you reach the point (0, 1) . (The unit circle)

### Sine and cosine as coordinates

- Example: What is sin(2π) ?
- Answer: 0
- Solution: The number 2π reaches the point (1, 0) . The sine is its y coordinate: sin(2π) = 0 . (The unit circle)

### Identities from the unit circle

- Example: You are told that sin x = −0.8 . What is sin(−x) ?
- Answer: 0.8
- Solution: The point of −x is the mirror image of the point of x in the x axis, so the y coordinate changes sign: sin(−x) = −sin x = 0.8 .

### Special values

- Example: What is sin(π/6) ?
- Answer: 1/2
- Solution: The point of π/6 on the unit circle is (√3/2, 1/2) , and the sine is its y coordinate: sin(π/6) = 1/2 . (The unit circle)

### The graphs of sine and cosine

- Example: What is the largest value of the function f(x) = −3 sin x ?
- Answer: 3
- Solution: The sine moves between −1 and 1 , so −3 sin x moves between −3 and 3 . The largest value is 3 .

### The tangent

- Example: What is tan(0) ?
- Answer: 0
- Solution: Divide the sine by the cosine: tan(0) = sin(0) / cos(0) = 0 / 1 = 0 .

### When there is a solution

- Example: Does the equation sin x = 0.9 have a solution?
- Options: No · Yes
- Answer: Yes
- Solution: The sine is always between −1 and 1 . The number 0.9 is in that range, so there is a solution.

### Trigonometric equations

- Example: Solve the equation sin x = √3/2 for 0 ≤ x ≤ π/2 .
- Answer: π/3
- Solution: There is one such number in the first quadrant, because sin(π/3) = √3/2 . The solution is x = π/3 .

### A periodic model

- Example: The height of a seat on a big wheel, in metres, after t minutes is h(t) = 14 + 10 sin(πt / 12) . What is the greatest height of the seat?
- Answer: 24 m
- Solution: The largest value of the sine is 1 , so h = 14 + 10 × 1 = 24 m.

## Not on the site yet

- Proving the identities
- General periodic functions

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).

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