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Cambridge IGCSE0580

Trigonometric functions

Mathematics 0580 Chapter Notes

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Trigonometric functions
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Trigonometric functions notes

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In this chapter

  1. 1The sine graph
  2. 2The cosine graph
  3. 3The tangent graph
  4. 4Rearrange, then use the inverse
  5. 5Finding the second solution
  6. 6Exact values without a calculator

1. The sine graph

The graph of y = sin x for 0° leqslant x leqslant 360° is a smooth wave, not straight lines. It starts at (0, 0) and rises to a maximum of 1 at x = 90°.

It crosses the x-axis at x = 180°, falls to a minimum of -1 at x = 270°, and returns to 0 at x = 360°.

-1 leqslant sin x leqslant 1

Examiner insight

“Some gained partial credit when their sketch strayed outside the limits of period or amplitude or when a negative sine curve was drawn.” “Infrequent errors seen were cosine or tangent sketches.”

Worked example 12 marks

Write down the coordinates of the maximum point and the minimum point on the graph of y = sin x for 0° leqslant x leqslant 360°.

  1. 1

    maximum point: (90, 1) [1]

  2. 2

    minimum point: (270, -1) [1]

2. The cosine graph

The graph of y = cos x is the same smooth wave, but it starts at (0, 1), the maximum, not at (0, 0).

It crosses the x-axis at (90, 0) and (270, 0), reaches its minimum of -1 at x = 180°, and ends at (360, 1). The curve has turning points at 0° and 360°, so it levels off there and is never a straight line.

-1 leqslant cos x leqslant 1

Examiner insight

“Most candidates plotted the main points of (0, 1), (90, 0), (180, –1), (270, 0) and (360, 1) but few knew the true shape of the curve and most were too linear, with some using a ruler to draw two straight lines joining these points.” “Some candidates incorrectly started their curve at (0, 0) rather than (0, 1).”

Worked example 12 marks

Sketch the graph of y = cos x for 0° leqslant x leqslant 360°.

  1. 1

    mark the points (0, 1), (90, 0), (270, 0) and (360, 1) [1]

  2. 2

    join them with a smooth curve through the minimum (180, -1) [1]

3. The tangent graph

The tangent graph is different. Since tan x = sin x ÷ cos x, it is undefined where cos x = 0. That gives asymptotes at x = 90° and x = 270°.

The graph crosses the x-axis at 0°, 180° and 360°. It rises towards each asymptote from the left and comes up from below on the right. The shape repeats every 180°.

tan x = (sin x)/(cos x)

Examiner insight

“Some attempts at the correct graph only covered part of the domain between the asymptotes or showed incorrect curvature.” “Some candidates sketched graphs of y = sin x or y = cos x.”

Worked example 12 marks

Write down the equations of the asymptotes of the graph of y = tan x for 0° leqslant x leqslant 360°.

  1. 1

    tan x = (sin x)/(cosx) is undefined where cos x = 0 [1]

  2. 2

    x = 90 and x = 270 [1]

4. Rearrange, then use the inverse

To solve a trigonometric equation, first make sin x, cos x or tan x the subject. Keep the sign: if the equation gives cos x = -0.4, the minus stays.

Then use the inverse function on the calculator. Do not round the value of sin x, cos x or tan x first. Give angles to 1 decimal place unless the question says otherwise.

A reflex angle is between 180° and 360°.

Examiner insight

“Some made errors with the rearrangement of the equation, losing the negative sign, which led to the two common answers of 66.4 and 293.6.” “There were also issues with premature rounding of figures leading to inaccurate final answers.”

Worked example 12 marks

cos x = 0.41

Find the reflex angle x.

  1. 1

    principal value: cos⁻¹ 0.41 = 65.79…° [1]

  2. 2

    reflex angle: x = 360° - 65.79…° = 294.2° [1]

5. Finding the second solution

A range of 0° to 360° usually holds two solutions. Sketch the graph, draw a horizontal line at the value, and read the second solution from the symmetry of the curve.

Let a be the angle the calculator gives. A negative angle is brought into range by adding 360°. The sine pair adds to 180° and the cosine pair adds to 360°, not the other way round.

sin x: x = a and 180° - a

cos x: x = a and 360° - a

tan x: x = a and a + 180°

Examiner insight

“Many candidates gave the solution x = 30° only and did not use their sketch to help identify the second solution of x = 210°.” “Candidates should consider how the sketch in part (a) can help in finding the location of further roots of the trigonometric equation.”

Worked example 13 marks

Solve the equation 3cos x + 1 = 0 for 0° leqslant x leqslant 360°.

  1. 1

    3cos x = -1, so cos x = -1/3 [1]

  2. 2

    x = cos⁻¹(-1/3) = 109.5° [1]

  3. 3

    second solution: x = 360° - 109.5° = 250.5° [1]

6. Exact values without a calculator

On a non-calculator paper the angle is a standard one. sin 60° = (√3)/2, so sin x = (√3)/2 gives x = 60° and 120°, which add to 180°.

For a negative value, start from the positive one: cos 60° = 1/2. On the cosine graph the curve is at -1/2 the same distance either side of 180°.

Examiner insight

“A common error was giving 330° (from 360° – 30°), indicating a misunderstanding of the symmetry properties of the sine function.”

Worked example 13 marks

Do not use a calculator.

Solve 2cos x + 1 = 0 for 0° leqslant x leqslant 360°.

  1. 1

    2cos x = -1, so cos x = -1/2 [1]

  2. 2

    cos 60° = 1/2, so from the graph x = 180° - 60° = 120° [1]

  3. 3

    second solution: x = 180° + 60° = 240° [1]

Recap

  • Sine and cosine waves stay between -1 and 1. Sine starts at (0, 0); cosine starts at (0, 1).
  • Cosine crosses the x-axis at 90° and 270°. Sine crosses at 0°, 180° and 360°.
  • Tangent has asymptotes at x = 90° and x = 270° and repeats every 180°.
  • Rearrange first and keep the sign. Use the inverse, then find the second solution.
  • Sine pair adds to 180°. Cosine pair adds to 360°. Tangent pair is 180° apart.
  • A negative calculator angle is not an answer. Add 360°, or 180° for tangent, until it is in range.
  • Give angles to 1 decimal place. Sketch the graph to see how many solutions there are.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write down the greatest value of sin x, and the value of x between 0° and 360° at which it occurs.2 marks
  2. Do not use a calculator. Write down the two values of x between 0° and 360° for which cos x = 0.2 marks
  3. A curve is one of y = sin x, y = cos x and y = tan x, drawn for 0° leqslant x leqslant 360°. It passes through (0, 0) and (180, 0) and has an asymptote at x = 90. State which graph it is, and write down the equation of its other asymptote.2 marks
  4. Do not use a calculator. Solve sin x = 0 for 0° leqslant x leqslant 360°.3 marks
  5. Solve 4sin x - 1 = 0 for 0° leqslant x leqslant 360°. Give your answers correct to 1 decimal place.3 marks
  6. Solve 7cos x + 3 = 0 for 0° leqslant x leqslant 360°. Give your answers correct to 1 decimal place.3 marks
  7. Do not use a calculator. Solve 2sin x - √3 = 0 for 0° leqslant x leqslant 360°.3 marks
  8. Solve 3tan x + 7 = 0 for 0° leqslant x leqslant 360°. Give your answers correct to 1 decimal place.3 marks
  9. Do not use a calculator. A seat is fixed to a big wheel. When the wheel has turned through x°, the height of the seat above the ground is h metres, where h = 5 + 4sin x for 0° leqslant x leqslant 360°. (a) Write down the greatest height of the seat and the value of x at which it occurs. [2] (b) Find the two values of x for which h = 3. [3] (c) Find the fraction of one full turn for which the seat is lower than 3 m. Give your answer in its simplest form. [2]7 marks
  10. The depth of water in a harbour, d metres, is modelled by d = 6.2 + 3.5cos x for 0° leqslant x leqslant 360°. The value of x is 0 at midnight and increases by 30 every hour. (a) Write down the least depth of the water and the value of x at which it occurs. [2] (b) Find the two values of x for which d = 5. Give your answers correct to 1 decimal place. [3] (c) A boat can use the harbour only when the depth is at least 5 m. Find the length of time, in hours correct to 1 decimal place, for which the boat cannot use the harbour. [2]7 marks

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