Ext Maths · 6 Trigonometry

6.4 Trigonometric functions

Before you start

The exact values of sin, cos and tan at $0 °$ , $30 °$ , $45 °$ , $60 °$ and $90 °$ (6.3).
Using the inverse keys $\sin^{- 1}$ , $\cos^{- 1}$ and $\tan^{- 1}$ in degree mode (6.2).
Rearranging a linear equation such as $5 y - 2 = 0$ .

Syllabus objectives covered (2)

Recognise, sketch and interpret the following graphs for 0° ⩽ x ⩽ 360°: • y = sin x • y = cos x • y = tan x.
Solve trigonometric equations involving sin x, cos x or tan x, for 0° ⩽ x ⩽ 360°.

Part 1 of 6

Sin, cos and tan beyond 90°

In a right-angled triangle every angle you use is acute, so SOH CAH TOA only gives values from $0 °$ to $90 °$ .
Your calculator gives values for any angle, though. Try $\sin 150 °$ : it gives $0.5$ , the same as $\sin 30 °$ .
Plot $y = \sin x$ , $y = \cos x$ and $y = \tan x$ from $0 °$ to $360 °$ and each makes a smooth repeating shape. Those shapes are the graphs this lesson is about.
Sine starts at 0, cosine starts at 1, and tangent has breaks at 90° and 270°.
The shapes are worth knowing well. With them you can predict how many answers an equation has and where the second answer lies.

Part 2 of 6

The sine and cosine graphs

** $y = \sin x$ ** is a wave. It starts at $0$ , rises to a maximum of $1$ at $x = 90 °$ , comes back to $0$ at $180 °$ , falls to a minimum of $- 1$ at $270 °$ , and returns to $0$ at $360 °$ .
** $y = \cos x$ ** is the same wave shifted. It starts at its maximum, $1$ at $0 °$ . It crosses $0$ at $90 °$ , reaches $- 1$ at $180 °$ , crosses $0$ again at $270 °$ and is back at $1$ at $360 °$ .
Both stay between $- 1$ and $1$ . So $\sin x = 1.5$ or $\cos x = - 2$ has no solutions.
Both repeat every $360 °$ . Between $0 °$ and $360 °$ you see exactly one full wave of each.
**The sine graph is symmetrical about $x = 90 °$ .** Any height on the first hump is reached at equal distances either side of $90 °$ . So:
$\sin (180 ° - x) = \sin x$
**The cosine graph is symmetrical about $x = 180 °$ .** So:
$\cos (360 ° - x) = \cos x$
These two symmetries are the whole secret of finding the second solution to an equation.

Part 3 of 6

The tangent graph

** $y = \tan x$ ** looks nothing like the other two. It passes through $0$ at $0 °$ , $180 °$ and $360 °$ .
It is $1$ at $45 °$ and shoots upwards as $x$ approaches $90 °$ . At $90 °$ there is no value, because $\tan 90 °$ doesn't exist (6.3).
The vertical line $x = 90 °$ is an asymptote: a line the graph gets closer and closer to but never touches. There is another at $x = 270 °$ . Draw them dashed.
Just after $90 °$ the graph reappears from far below, rises through $- 1$ at $135 °$ and $0$ at $180 °$ , and repeats the same shape.
Tan repeats every $180 °$ , not every $360 °$ . So:
$\tan (180 ° + x) = \tan x$
Unlike sine and cosine, tan has no maximum or minimum. It takes every value, however large or small.

Part 4 of 6

Sketching and interpreting

A sketch is not a plot. You don't need a table of values, but it must show the key features in the right places.
For sine or cosine: label the $x$ -axis at $90 °$ , $180 °$ , $270 °$ and $360 °$ , mark $1$ and $- 1$ on the $y$ -axis, and draw a smooth curve through the correct zeros, maximum and minimum. No pointed peaks, no straight lines.
For tangent: mark $0 °$ , $180 °$ and $360 °$ where it crosses, draw the asymptotes at $90 °$ and $270 °$ as dashed lines, and show three branches that bend towards them.
Interpreting mostly means reading off how many times a horizontal line $y = k$ cuts the curve. Each crossing is one solution of $\sin x = k$ (or $\cos x = k$ , or $\tan x = k$ ).
For $\sin x = k$ : two solutions if $- 1 < k < 1$ but $k \neq 0$ . Three if $k = 0$ ( $0 °$ , $180 °$ , $360 °$ ). One if $k = 1$ or $k = - 1$ . None if $k$ is outside $- 1$ to $1$ .
For $\cos x = k$ : two solutions for any $k$ from $- 1$ up to $1$ except $k = - 1$ , which has one ( $180 °$ ). Note $\cos x = 1$ has two: $0 °$ and $360 °$ .
For $\tan x = k$ : two solutions for any $k \neq 0$ , and three when $k = 0$ .

Part 5 of 6

Solving an equation: get to the first solution

Step 1: rearrange so that $\sin x$ , $\cos x$ or $\tan x$ is on its own. For example, $5 \cos x - 2 = 0$ becomes $\cos x = 0.4$ . Treat $\cos x$ like a single unknown.
Step 2: check the value is possible. If it is $\sin x$ or $\cos x$ and the number is bigger than $1$ or less than $- 1$ , stop: there are no solutions.
Step 3: find the first solution. If the value is an exact value from 6.3, write the angle down. Otherwise use the inverse key.
The calculator gives only one angle, and it may be outside $0 °$ to $360 °$ . $\sin^{- 1}$ and $\tan^{- 1}$ give answers between $- 90 °$ and $90 °$ , and $\cos^{- 1}$ gives answers between $0 °$ and $180 °$ .
Keep that first value unrounded on the calculator while you find the others.

Part 6 of 6

Solving an equation: find every solution in the range

Use the symmetry of the graph to get the other solutions. Call the calculator's answer $α$ .
For sine the pair is balanced about 90°, so the second answer is 180° − α. For cosine it is balanced about 180°, so the second answer is 360° − α.
Sine: the solutions are $α$ and $180 ° - α$ . For $\sin x = 0.4$ : $α = 23.6 °$ , so $x = 23.6 °$ or $180 ° - 23.6 ° = 156.4 °$ .
Cosine: the solutions are $α$ and $360 ° - α$ . For $\cos x = 0.4$ : $x = 66.4 °$ or $360 ° - 66.4 ° = 293.6 °$ .
Tangent: the solutions are $α$ and $α + 180 °$ . For example, $\tan x = \sqrt{3}$ gives $x = 60 °$ or $240 °$ .
When the calculator gives a negative angle (from $\sin^{- 1}$ or $\tan^{- 1}$ of a negative number), it is outside the range. Don't give it as an answer.
For sine with negative $α$ , use $180 ° - α$ and $360 ° + α$ . For tangent with negative $α$ , use $α + 180 °$ and $α + 360 °$ .
Finish with a check. Both answers must be between $0 °$ and $360 °$ . Put each one back into the original equation on your calculator to confirm.
The trap is stopping after the calculator's answer. On a 3-mark equation that loses at least a mark, and often the answer the question was really testing.
Give non-exact angles to one decimal place.

Worked examples

Example 1. Without a calculator, solve $2 \sin x - 1 = 0$ for $0 ° ⩽ x ⩽ 360 °$ .

Show the working

Rearrange to get $\sin x$ on its own: add $1$ , then divide by $2$ .
$\sin x = \frac{1}{2}$
$\frac{1}{2}$ is an exact value: $\sin 30 ° = \frac{1}{2}$ . So one solution is $30 °$ .
Sine is symmetrical about $90 °$ , so the second solution is $180 ° - 30 °$ .
$x = 180 ° - 30 ° = 150 °$
Check: $\frac{1}{2}$ is positive, so the answers lie where the sine graph is above the axis, between $0 °$ and $180 °$ . Both do.

Answer: $x = 30 °$ or $x = 150 °$ .

Example 2. Solve $5 \cos x - 2 = 0$ for $0 ° ⩽ x ⩽ 360 °$ .

Show the working

Rearrange.
$\cos x = \frac{2}{5} = 0.4$
$0.4$ is between $- 1$ and $1$ , so solutions exist. Use the inverse key.
$α = \cos^{- 1} (0.4) = 66.421 \dots °$
Cosine is symmetrical about $180 °$ , so the second solution is $360 ° - α$ .
$360 ° - 66.421 \dots ° = 293.578 \dots °$
Round each to one decimal place.

Answer: $x = 66.4 °$ or $x = 293.6 °$ .

Example 3. Solve $10 \sin x + 3 = 0$ for $0 ° ⩽ x ⩽ 360 °$ .

Show the working

Rearrange.
$\sin x = - 0.3$
The calculator gives a negative angle, which is outside the range.
$α = \sin^{- 1} (- 0.3) = - 17.457 \dots °$
Sine is negative between $180 °$ and $360 °$ (the part of the wave below the axis), so both answers are there.
Use $180 ° - α$ and $360 ° + α$ .
$180 ° - (- 17.457 \dots °) = 197.457 \dots ° 360 ° + (- 17.457 \dots °) = 342.542 \dots °$
Round to one decimal place. Check on the calculator: $\sin 197.5 °$ and $\sin 342.5 °$ both give about $- 0.3$ .

Answer: $x = 197.5 °$ or $x = 342.5 °$ .

Example 4. Solve $3 \tan x + 7 = 0$ for $0 ° ⩽ x ⩽ 360 °$ .

Show the working

Rearrange.
$\tan x = - \frac{7}{3}$
Tan takes every value, so there are solutions. The calculator gives a negative angle.
$α = \tan^{- 1} (- \frac{7}{3}) = - 66.801 \dots °$
Tan repeats every $180 °$ , so add $180 °$ and then another $180 °$ until you land in the range.
$- 66.801 \dots ° + 180 ° = 113.198 \dots ° - 66.801 \dots ° + 360 ° = 293.198 \dots °$
Check with the graph: tan is negative on the branches between $90 °$ and $180 °$ , and between $270 °$ and $360 °$ . One answer is in each.

Answer: $x = 113.2 °$ or $x = 293.2 °$ .

Where people go wrong

Mistake: Giving only the calculator's answer.

Instead: Use the graph's symmetry to find every solution between $0 °$ and $360 °$ .

The inverse key returns one angle by design, but the graph crosses the line $y = k$ more than once in the range.

Mistake: Giving a negative angle such as $- 17.5 °$ as a solution.

Instead: Convert it: for sine use $180 ° - α$ and $360 ° + α$ ; for tangent add $180 °$ and $360 °$ .

The question restricts answers to $0 ° ⩽ x ⩽ 360 °$ , and a negative angle is outside that.

Mistake: Using $180 ° - α$ for cosine.

Instead: For cosine the partner is $360 ° - α$ .

The cosine graph is symmetrical about $180 °$ , not $90 °$ . Check: $\cos 66.4 °$ is positive, but $\cos 113.6 °$ is negative.

Mistake: Drawing the tan graph as a continuous curve through $90 °$ .

Instead: Draw dashed asymptotes at $90 °$ and $270 °$ , with separate branches.

$\tan 90 °$ and $\tan 270 °$ don't exist, and the graph jumps from very large to very negative at those lines.

Mistake: Splitting the $x$ away from cos when rearranging, as in turning $5 \cos x = 2$ into $\cos = \frac{2}{5 x}$ .

Instead: Treat $\cos x$ as one unknown and divide by $5$ only: $\cos x = \frac{2}{5}$ .

$\cos x$ is a single quantity, the cosine of the angle $x$ . 'cos' on its own means nothing, so the $x$ has to stay attached to it.

Check yourself

Write down the values of $x$ between $0 °$ and $360 °$ where $\sin x = 0$ .
Answer
$0 °$ , $180 °$ and $360 °$ .
State the minimum value of $\cos x$ for $0 ° ⩽ x ⩽ 360 °$ , and where it happens.
Answer
$- 1$ , at $x = 180 °$ .
Write down the equations of the asymptotes of $y = \tan x$ for $0 ° ⩽ x ⩽ 360 °$ .
Answer
$x = 90 °$ and $x = 270 °$ .
Explain why $\sin x = 1.2$ has no solutions.
Answer
The sine graph never goes above $1$ or below $- 1$ , so the line $y = 1.2$ never meets it.
One solution of $\sin x = 0.4$ is $23.6 °$ . Write down the other solution between $0 °$ and $360 °$ .
Answer
$180 ° - 23.6 ° = 156.4 °$ .
One solution of $\cos x = 0.4$ is $66.4 °$ . Write down the other solution between $0 °$ and $360 °$ .
Answer
$360 ° - 66.4 ° = 293.6 °$ .
Without a calculator, solve $\tan x = \sqrt{3}$ for $0 ° ⩽ x ⩽ 360 °$ .
Answer
$\tan 60 ° = \sqrt{3}$ , so $x = 60 °$ or $60 ° + 180 ° = 240 °$ .
How many solutions does $\cos x = - 1$ have for $0 ° ⩽ x ⩽ 360 °$ ?
Answer
One: $x = 180 °$ , the minimum of the graph.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. Do not use a calculator in this question.

(a) Sketch the graph of $y = \cos x$ for $0 ° ⩽ x ⩽ 360 °$ . 2 marks No calculator

Mark scheme

B1 one complete cosine wave, starting and ending at a maximum, with a smooth shape
B1 passes through (90, 0), (180, −1) and (270, 0), with 1 and −1 indicated on the y-axis

(b) Solve $\cos x = \frac{1}{2}$ for $0 ° ⩽ x ⩽ 360 °$ . 2 marks No calculator

Mark scheme

B1 60
B1 300

(c) Write down the number of solutions of $\cos x = - 0.9$ for $0 ° ⩽ x ⩽ 360 °$ . 1 mark No calculator

Mark scheme

B1 2 cao

Question 2.

(a) Solve $3 \cos x + 2 = 0$ for $0 ° ⩽ x ⩽ 360 °$ . 3 marks

Mark scheme

M1 cos x = −2/3 oe
A1 131.8 awrt
A1 228.2 awrt; FT 360 − their first angle

Question 3.

(a) Solve $2 \tan x = 5$ for $0 ° ⩽ x ⩽ 360 °$ . 3 marks

Mark scheme

M1 tan x = 2.5 oe
A1 68.2 awrt
A1 248.2 awrt; FT their first angle + 180

Question 4. Do not use a calculator in this question.

(a) Write down the equations of the two asymptotes of $y = \tan x$ for $0 ° ⩽ x ⩽ 360 °$ . 2 marks No calculator

Mark scheme

B1 x = 90
B1 x = 270

(b) Explain why the equation $\sin x = 1.5$ has no solutions. 1 mark No calculator

Mark scheme

B1 sin x can only take values from −1 to 1 (the graph never goes above 1) oe

(c) $\sin 40 ° = 0.643$ , correct to 3 significant figures. Write down another angle between $0 °$ and $360 °$ whose sine is $0.643$ , correct to 3 significant figures. 1 mark No calculator

Mark scheme

B1 140 cao

How this is examined

Solving a trig equation and finding both answers in the range: A 3-mark Solve on the calculator paper, such as $3 \cos x + 2 = 0$ for $0 ° ⩽ x ⩽ 360 °$ . The non-calculator version uses an exact value. M1 for rearranging to $\cos x = - \frac{2}{3}$ . A1 for the first angle to 1 d.p. A1 for the second angle, which is usually allowed to follow through from a wrong first angle if the symmetry rule is right, such as $360 ° - α$ .
Sketching sin, cos or tan with the key features in the right places: A 2-mark Sketch on given axes from $0 °$ to $360 °$ . One mark for the correct shape over the full range. One for the key points: zeros, maximum $1$ and minimum $- 1$ at the right angles, or the asymptotes at $90 °$ and $270 °$ for tan.
Reading the number of solutions, or a second solution, from the graph: A 1-mark Write down after a sketch, or a question that gives one solution and asks for another. The correct count or angle, using the graph's symmetry, such as $180 ° - 40 ° = 140 °$ for sine.
Explaining why an equation has no solutions: A 1-mark Explain. Saying that sine or cosine only takes values from $- 1$ to $1$ , so the value given is impossible.

Words to know

Asymptote: A line that a graph gets closer and closer to but never meets. $y = \tan x$ has asymptotes at $x = 90 °$ and $x = 270 °$ .
Period: The width of one repeat of a graph: $360 °$ for sine and cosine, $180 °$ for tangent.
Principal value: The single angle the calculator's inverse key gives; the other solutions come from the graph's symmetry.
Sketch: A simple freehand drawing showing the key features of a graph, not an accurate plot.

All Ext Maths lessons Next: 6.5 Non-right-angled triangles