The graphs of y = sin θ and y = cos θ both have amplitude 1 and period 360°, while y = tan θ has period 180° and vertical asymptotes at 90°, 270° and so on. Transformations stretch, shift and reflect these curves in predictable ways, which lets you read an equation from a graph, find maximum and minimum points, and count how many solutions an equation has.
Key features of the trig graphs
- y = sin θ: amplitude 1, period 360°, passes through the origin
- y = cos θ: amplitude 1, period 360°, maximum at (0°, 1)
- y = tan θ: period 180°, no amplitude, asymptotes at θ = 90°, 270°, …
- y = a sin(kθ) + b: amplitude |a|, period 360° / k, centre line y = b
What do the sin, cos and tan graphs look like?
Sine and cosine are the same wave shifted by 90°. Both run between −1 and 1 and repeat every 360°, but cosine starts at its maximum at θ = 0° while sine starts at the origin. Tangent is different in kind: it has no maximum or minimum, it repeats every 180°, and it has vertical asymptotes where cos θ = 0.
That 180° period matters when you count solutions, because tan θ = k has a solution in every 180°, not every 360°.
How do transformations change the trig graphs?
The usual graph transformations apply:
- y = a f(θ) stretches vertically, so the amplitude is multiplied by a
- y = f(θ) + a moves the graph up by a
- y = f(θ − a) moves the graph a to the right
- y = f(aθ) squashes it horizontally, dividing the period by a
- y = −f(θ) reflects it in the horizontal axis
Tangent has no amplitude to stretch. Instead, a horizontal stretch or a translation moves its asymptotes. To track a particular point, apply the change to its coordinates: horizontal changes act on θ, vertical changes act on y.
How do you find the equation from a graph?
For y = A sin θ + B or y = A cos θ + B, the maximum and minimum give both constants: A = (max − min) / 2 and B = (max + min) / 2. In words, half the gap between them is the amplitude, and the midpoint is the vertical shift.
The period gives the coefficient of θ: for y = A sin(kθ) + B the period is 360° / k, and for tangent it is 180° / k. A tangent graph has no amplitude to read, so use the spacing of its asymptotes instead.
How many solutions does a trig equation have?
Imagine the line y = c drawn across the curve and count the crossings. Find how many full periods fit into the interval, then how many times the line crosses one period, and multiply. sin(kθ) = c fits k periods into 0° to 360°, so it has k times as many solutions as sin θ = c.
Tangent crosses a horizontal line only once per period, so the count works differently for it.
Working with a letter instead of a number
Some questions give a point an unknown coordinate, p, and ask you to work with that. Symmetry relates values: if sin α = p, then sin(180° − α) = p as well; but if cos α = p, then cos(180° − α) = −p. A transformation acts on p exactly as it acts on y, so under y = 2 sin θ − 1 the value becomes 2p − 1. These questions often ask for the possible values of p too, read from the graph.
Worked examples
Example 1. The curve y = cos θ is drawn for 0° ≤ θ ≤ 360°. Write down its maximum point and the number of times it crosses the θ-axis.
- Cosine starts at its peak: the maximum is (0°, 1). (It reaches 1 again at 360°.)
- It crosses the axis at θ = 90° and θ = 270°, so twice.
Answer: (0°, 1); 2 crossings.
Marks: B1 for the maximum, B1 for the crossings.
Example 2. Find the maximum and minimum values of y = 2 cos θ + 1, and the image of the maximum point (0°, 1) of y = cos θ.
- The vertical stretch by 2 makes the values run from −2 to 2.
- The translation up by 1 moves them to −1 to 3.
- The point (0°, 1) keeps θ = 0° and its y becomes 2(1) + 1 = 3.
Answer: Maximum 3, minimum −1; the image is (0°, 3).
Marks: M1 for the stretch, A1 for the maximum and minimum, B1 for the image.
Example 3. The curve y = 3 sin θ + k has a maximum y-value of 10. Find k, and the minimum point with the smallest positive θ-coordinate.
- 3 sin θ peaks at 3, so 3 + k = 10 and k = 7.
- 3 sin θ has its lowest value −3, so the minimum of the curve is −3 + 7 = 4.
- Sine is at its minimum when θ = 270°.
Answer: k = 7; the minimum point is (270°, 4).
Marks: M1 for 3 + k = 10, A1 for k = 7, A1 for the point.
Example 4. Given that 0 < c < 1, how many solutions does sin 3θ = c have for 0° ≤ θ ≤ 360°?
- y = sin 3θ has period 360° / 3 = 120°, so three full waves fit into the interval.
- The line y = c crosses each wave twice.
- Total: 3 × 2 = 6 solutions.
Answer: 6 solutions.
Marks: M1 for three periods, A1 for 6.
Common mistakes
- Using 360° as the period of tan. tan θ repeats every 180°, so its solutions and asymptotes are 180° apart.
- Moving the graph the wrong way. y = f(θ − 30°) moves the curve 30° to the right, not to the left.
- Multiplying the period instead of dividing it. y = sin 2θ has period 180°, half of 360°, because the wave runs twice as fast.
- Confusing amplitude with the maximum. For y = 3 sin θ + 7, the amplitude is 3 but the maximum value is 10.
- Assuming symmetry always keeps the sign. sin(180° − α) = sin α, but cos(180° − α) = −cos α.
How trig graphs are examined
In Edexcel International A-Level, trig graphs are Pure 1 content (paper WMA11), within the trigonometric ratios chapter; Edexcel A-Level (9MA0) covers them in Year 1. Expect questions that ask you to:
- state the coordinates of maximum or minimum points, or the positions of asymptotes
- find an unknown constant from the maximum or minimum of a transformed curve
- count the solutions of an equation over an interval
- express values in terms of a letter, such as p, using symmetry
Questions in exactly this style, with mark schemes, are in the Pure 1 trigonometric ratios exam questions. The trigonometric ratios worksheet and the trigonometry revision engine give extra practice.
Transforming trig graphs uses the same rules as transformations of graphs, and it underpins solving trig equations in Pure 2.
Frequently asked questions
What is the period of tan x?
180°, or π in radians. The tangent graph repeats every 180°, with an asymptote every 180° as well.
What is the amplitude of y = 4 cos θ?
4. The curve runs from −4 to 4. Tangent graphs have no amplitude because they have no maximum or minimum.
How does y = sin 2θ differ from y = 2 sin θ?
y = sin 2θ is squashed horizontally, so its period halves to 180°. y = 2 sin θ is stretched vertically, so its amplitude doubles to 2.