Trigonometry
Grade 7-8Trigonometric Graphs
The graphs of sine and cosine repeat the same wave shape forever, and a handful of key features, the period, the amplitude and the maximum and minimum points, describe everything about them. This lesson covers the shape and key points of y = sin x° and y = cos x°, vertical translations, vertical stretches, and using these graphs to model real-world periodic situations.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
The shape and key points of sine and cosine graphs
Both y = sin x° and y = cos x° repeat every 360°, which is called the period, and both oscillate between a maximum of 1 and a minimum of -1, giving an amplitude of 1. The graph of y = sin x° starts at (0, 0), rises to a maximum of (90°, 1), returns to 0 at 180°, falls to a minimum of (270°, -1), and returns to 0 at 360°. The graph of y = cos x° starts at a maximum of (0°, 1), falls to 0 at 90°, reaches a minimum of (180°, -1), returns to 0 at 270°, and rises back to (360°, 1).
The graph of y = sin x° crosses the x-axis at 0°, 180° and 360°, while the graph of y = cos x° crosses the x-axis at 90° and 270°.
Vertical translations
Adding a number outside the trig function, as in y = sin x° + a, shifts the whole graph vertically by a, without changing its shape, period or amplitude. Every key point, including the maximum and minimum, moves up or down by the same amount.
The graph of y = sin x° + 2 has the same shape as y = sin x°, but shifted up by 2, so its maximum becomes 1 + 2 = 3 and its minimum becomes -1 + 2 = 1.
Vertical stretches
Multiplying the trig function by a number, as in y = a sin x°, stretches the graph vertically by scale factor a, changing the amplitude to a, so the maximum becomes a and the minimum becomes -a. The period and the x-values of the key points stay the same.
The graph of y = 3 cos x° has amplitude 3, so its maximum is 3 and its minimum is -3, but it still has a period of 360° and still peaks at x = 0°.
Modelling real-world periodic situations
A model in the form y = a + b sin x° (or cos x°) combines a vertical translation and a vertical stretch: a is the fixed baseline value, and b is the amplitude of the variation around it. The maximum value is a + b, and the minimum value is a - b.
The height of a car on a Ferris wheel is modelled by h = 20 + 18 sin x°. The maximum height is 20 + 18 = 38 m, and the minimum height is 20 - 18 = 2 m.
Worked Examples
Three exam-style questions, fully solved.
Write down the period of the graph of y = sin x°.
Easy- 1.Recall how often the sine graph repeats its pattern
Answer: 360°
The graph shown has the same shape as y = sin x°, but shifted vertically up by 2. Write down the equation of the graph.
Medium- 1.Recognise that a vertical shift adds a constant outside the trig function
- 2.Add the shift to the original function: sin x° + 2
Answer: y = sin x° + 2
The height, h metres, of a car on a Ferris wheel above the ground is modelled by h = 20 + 18 sin x°, where x° is the angle turned from the start. Write down the maximum height and the minimum height of the car.
Hard- 1.Find the maximum by adding the amplitude to the baseline: 20 + 18 × 1
- 2.Find the minimum by subtracting the amplitude from the baseline: 20 + 18 × (-1)
Answer: Maximum height 38 m, minimum height 2 m
Avoid These
The most common mistakes students make.
Forgetting that the period of both sin x° and cos x° is 360°, and misreading how often a graph completes one full cycle.
Confusing a vertical translation, y = sin x° + a, which shifts the whole graph up or down, with a vertical stretch, y = a sin x°, which changes the amplitude.
Mixing up the key points of sine and cosine, for example forgetting that sine peaks at 90° while cosine peaks at 0°.
In modelling questions, forgetting that the minimum value uses subtraction (baseline minus amplitude), not addition.
Giving only the y-value of a maximum or minimum point without also stating the corresponding value of x, when both are asked for.
FAQ
Questions parents and students ask.
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