1. The three graphs
Why the graphs matter
- Beyond a right-angled triangle the angle is no longer restricted to acute values, so sin and cos are defined for every angle from 0° to 360° and beyond, and tan for every angle except 90°, 270° and every 180° on from them
- The syllabus asks you to recognise, sketch and interpret y = sin x, y = cos x and y = tan x over 0° ⩽ x ⩽ 360°
- The graphs are what make an equation solvable, because a calculator returns only one answer and the graph reveals the rest
Sine and cosine
- Both are waves rising to 1 and falling to −1, and both repeat every 360°, which is their period
- Plotting at every multiple of 90° is enough to get the shape right
| x | 0° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|
| sin x | 0 | 1 | 0 | −1 | 0 |
| cos x | 1 | 0 | −1 | 0 | 1 |
- y = sin x passes through the origin, while y = cos x starts at its peak, crossing the vertical axis at 1
- The cosine curve is the sine curve shifted 90° to the left, so the two have the same shape in different positions
- Sine is symmetrical about x = 90° and cosine about x = 180°, and those symmetries are what generate second solutions
Tangent
- y = tan x breaks the pattern entirely: rather than one continuous wave it comes in separate branches, repeating every 180°, half the period of the other two
- Each branch climbs from far below the axis to far above it
- Vertical asymptotes sit at x = 90° and x = 270°, where the curve rises without ever reaching the line
- The curve passes through the origin, and through (180°, 0) and (360°, 0)
- Tangent has no value at 90° or 270°, which is why those are drawn as dotted lines rather than points
Worked example
Reading values off the graphs
Use the graphs to write down the value of sin 180°, cos 180° and the two values of x between 0° and 360° for which sin x = √2/2.
Solution:
- The sine curve starts at 0, rises to 1 at 90°, returns to 0 at 180°, falls to −1 at 270° and returns to 0 at 360°
- So sin 180° = 0
- The cosine curve starts at 1, falls to 0 at 90°, reaches −1 at 180°, returns to 0 at 270° and back to 1 at 360°
- So cos 180° = −1
- √2/2 is about 0.71, so draw the horizontal line y = 0.71 across the sine curve
- It cuts the curve twice between 0° and 360°, both times where the curve is above the axis
- √2/2 is the exact value of sin 45°, so the first is x = 45°, with no calculator needed
- The sine curve is symmetrical about x = 90°, so the second is 180 − 45 = 135°
- The two solutions are x = 45° and x = 135°
2. Solving a trigonometric equation
Why there is more than one answer
- A question gives an interval, almost always 0° ⩽ x ⩽ 360°, and expects every solution inside it
- The calculator's inverse function returns just one angle, so the others come from the symmetry of the graph
- Two answers is the usual number over a 360° interval for sine or cosine, and the answer line normally shows two spaces
- Tangent behaves differently: its solutions are 180° apart rather than mirrored
The method
- Rearrange first, so the equation reads sin x = …, cos x = … or tan x = …, treating sin x as if it were a single unknown
- Take the inverse function to get the first angle
- Sketch the relevant graph across the interval and draw the horizontal line at that value
- Read off where else the line cuts the curve, using the symmetry rules
- For sine, the second solution is 180° − the first
- For cosine, the second solution is 360° − the first
- For tangent, add 180° to move to the next branch
- Where the first angle comes out negative, apply the symmetry rule to it as it stands, then add 360° to any answer still below 0°; for tangent, keep adding 180° and keep every answer that lands inside the interval
- Check each answer by substituting it back
Worked example
A sine equation with a negative value
Solve 3 sin x + 2 = 0 for 0° ⩽ x ⩽ 360°. Give your answers correct to 1 decimal place.
Solution:
- Rearrange to isolate sin x: 3 sin x = −2, so sin x = −2 ÷ 3
- The calculator gives sin⁻¹(−0.6666…) = −41.8103…, which is outside the interval
- Sine is negative below the axis, so both solutions lie between 180° and 360°
- Using the symmetry about 270°, the solutions are 180° + 41.8103… and 360° − 41.8103…
- So x = 221.8° or x = 318.2° to 1 decimal place
- Check: sin 221.8° and sin 318.2° both give −0.667 to 3 decimal places
Worked example
A cosine equation
Solve 4 cos x = 1 for 0° ⩽ x ⩽ 360°. Give your answers correct to 1 decimal place.
Solution:
- Divide by 4: cos x = 0.25
- The calculator gives cos⁻¹(0.25) = 75.5224…
- For cosine the second solution is 360° minus the first
- 360 − 75.5224… = 284.4775…
- So x = 75.5° or x = 284.5° to 1 decimal place
- Check: the two answers are symmetrical about 180°, as the cosine curve requires
Worked example
A tangent equation without a calculator
Without a calculator, solve tan x + √3 = 0 for 0° ⩽ x ⩽ 360°.
Solution:
- Rearrange to isolate tan x: tan x = −√3
- There is no calculator, so use the exact values: tan 60° = √3
- Changing the sign of the angle changes the sign of tan, so tan(−60°) = −√3 and the first angle is −60°, which is outside the interval
- Add 180° to move to the next branch: −60 + 180 = 120°
- Add another 180°: 120 + 180 = 300°
- One more would give 480°, which is outside the interval, so there are no others
- So x = 120° or x = 300°
- Check: tangent is negative between 90° and 180° and between 270° and 360°, and one answer lies in each
Exam tip
Show the first angle and always give its partner
Write the first angle you find in your working, even when it is negative or is not the angle asked for: on its own it can earn a mark. Then always give its partner. A pair inside the interval that adds to 360° for cosine, to 180° for sine (540° if both are reflex) or differs by 180° for tangent can still earn a mark when the first angle is wrong, so check that your pair fits.
Reading a reflex answer
- A question may give the value and ask specifically for the reflex angle, meaning the one between 180° and 360°
- Work out the first angle (from the calculator, or from the exact values on the non-calculator paper), then apply the symmetry rule to reach the reflex partner
- This is the same method, simply asking for the second solution instead of both
3. Sketching in an answer
What a sketch must show
- A sketch is asked for in its own right, usually for 2 marks, and it needs the shape and the key points rather than plotted accuracy
- Mark where the curve meets the axes, and show the maximum of 1 and minimum of −1 for sine and cosine
- Run the curve across the whole interval the question states, no further and no less
- Draw it freehand and smoothly, never with a ruler
- For a tangent sketch, put the asymptotes in as dotted vertical lines, since the branches make no sense without them
- Where the sketch is part (a) and an equation is part (b), the sketch is meant to be used, so draw the horizontal line on it and mark the crossings
Worked example
Sketching a curve to place the answers
Sketch y = cos x for 0° ⩽ x ⩽ 360° and use it to explain why cos x = −0.7 has two solutions in that interval.
Solution:
- Mark the axes first: x from 0° to 360°, and y from −1 to 1, since the curve never leaves that range
- Plot the five points the curve is fixed by: (0, 1), (90, 0), (180, −1), (270, 0) and (360, 1)
- Join them with one smooth curve, drawn freehand rather than ruled
- Draw the horizontal line y = −0.7 across the sketch
- The line sits below the axis, and the curve is below the axis between 90° and 270°
- The line therefore crosses the curve twice, once on the way down and once on the way back up
- So there are two solutions, one between 90° and 180° and one between 180° and 270°
- The sketch is what shows where the second answer lives; the calculator only ever gives the first
