Chapter 5 of 8

All 8 chapters
  1. 1Function notation, readunderstood
  2. 2Composite and inverse functions, readunderstood
  3. 3Cubic, reciprocal and circle graphs, readunderstood
  4. 4Exponential graphs, readunderstood
  5. 5Graphs of sin, cos and tan, readunderstood
  6. 6Moving graphs, readunderstood
  7. 7Gradients and areas under graphs, readunderstood
  8. 8Solving with graphs and trials, readunderstood

M4.10.5 Chapter 5

Graphs of sin, cos and tan

What does the graph of y = sin x look like, for angles from 0° to 360°?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Sketch y = sin x and y = cos x up to 360°
  • Use a graph to find two angles with sin x = 0.5

In a right angled triangle, sin of an angle is a ratio of two sides. Here you use sin and cos for any angle from 0° to 360°. Check that your calculator is in degree mode. If sin 30° does not show 0.5, it is not.

Think of a wheel with radius 1. A point P sits on its rim, level with the centre, on the right. Turn the wheel anticlockwise by an angle x. Now P has a height above the centre line, and a distance across from the centre.

We define sin x as the height of P above the centre. We define cos x as its distance to the right of the centre. Below the centre the height is negative. To the left of the centre the distance is negative.

A graph with the axis across and the axis height, with equal units. A circle of radius 1 is drawn round the origin. A point P on the circle is at (0.87, 0.5), and a point Q is at (−0.87, 0.5). Dashed lines join the origin to P and to Q. Dashed lines also run from P and from Q across to the height axis at 0.5, and down to the across axis.PQ−0.50.5−0.50.50acrossheight
The wheel of radius 1. P is 30° round from the right, and Q is 150° round. Both are at height 0.5.

At 30° the point P is at height 0.5, so sin 30° = 0.5. At 150° the point Q is at the same height, so sin 150° = 0.5 too. As the wheel turns, the height of P rises and falls like a wave. Plot it against the angle, and you get the graph of y = sin x.

A graph with the axis across and the axis height, with equal units. A circle of radius 1 is drawn round the origin. A point P on the circle is at (−0.87, −0.5). A dashed line joins the origin to P, and dashed lines run from P to both axes, so the height of P is −0.5.P0.5−0.50.50acrossheight
At 210° the point is 30° below the centre line, on the left. Its height is −0.5.

Worked example

Find sin 210°, using the wheel.

  1. Work out how far 210° is past the half turn: 210° − 180° = 30°.
  2. So P is 30° below the horizontal line, on the left. Its distance below the centre is the same as the height at 30°, which is 0.5.
  3. Below the centre means negative. So sin 210° = −0.5.

Answer: sin 210° = −0.5.

A graph with the x axis x° and the y axis y. A smooth wave for y = sin x starts at the origin, rises to 1 at 90 degrees, crosses the x axis at 180 degrees, falls to −1 at 270 degrees, and returns to 0 at 360 degrees. Thirteen dots mark the curve at every 30 degrees.90180270360−1−0.50.510x°y
The graph of y = sin x, with x in degrees. Each dot is the height of the wheel point after that angle.

Read the graph from left to right. At 0° it is 0. It rises to 1 at 90° and falls to 0 at 180°. Then it reaches −1 at 270° and is back to 0 at 360°. After 360° the wheel is back where it began, so the wave repeats. The length of one full wave is its period: 360° for sin and cos.

The graph of cos x is the same wave, but it starts at the top. At 0° it is 1. It falls to 0 at 90°, to −1 at 180°, rises to 0 at 270°, and is back at 1 at 360°. Cos x is the distance of P across from the centre, so it starts at 1.

A graph with the x axis x° and the y axis y. A solid wave for y = sin x starts at 0 and rises. A dashed wave for y = cos x starts at 1 and falls. They cross at 45 degrees and at 225 degrees. Dots mark where the waves meet the x axis, at 90, 180, 270 and 360 degrees. A key under the graph names the solid wave y = sin x and the dashed wave y = cos x.90180270360−1−0.50.510x°yy = sin xy = cos x
The two waves. The cos wave is the sin wave moved 90° to the left.

Both waves stay between −1 and 1. So sin x = 2 has no solution at all. The graph of tan x is different. It is sin x divided by cos x. Cos x is 0 at 90° and 270°, so tan x has no value there.

A graph with the x axis x° and the y axis y. Dots mark the points (180, 0) and (360, 0) on the x axis. The curve for y = tan x has three separate branches. The first rises from 0 at the origin and shoots up towards a dashed vertical line at 90 degrees. The second comes up from far below that line, crosses the x axis at 180 degrees and climbs towards a dashed line at 270 degrees. The third comes up from far below that line to 0 at 360 degrees.90180270360−4−2240x°y
The graph of y = tan x. The dashed lines at 90° and 270° are asymptotes. The curve gets close but never touches them.

The tan graph passes through 0 at 0°, 180° and 360°. It repeats every 180°, so its period is 180°. At 45° it is 1. Between 90° and 180° it is negative.

Now use the graph to solve an equation. To solve sin x = 0.5, draw the horizontal line y = 0.5 across the sin graph. Each place where it meets the curve gives an angle.

Worked example

Solve sin x = 0.5 for x between 0° and 360°.

  1. Draw the line y = 0.5 across the graph of y = sin x.
  2. It meets the curve twice between 0° and 360°.
  3. The calculator gives the first angle: sin⁻¹ 0.5 = 30°.
  4. The wave is symmetric about 90°. The second angle is 180° − 30° = 150°.

Answer: x = 30° or x = 150°.

See it three ways

With things

A valve on a bicycle wheel starts level with the centre, on the right, and the wheel turns anticlockwise. At what angles is the valve half a radius above the centre?

As a picture

A graph with the x axis x° and the y axis y. A wave for y = sin x. A dashed horizontal line at height 0.5 meets the wave at two points, A at 30 degrees and B at 150 degrees.AB60120240300−1−0.50.510x°y

In words

The valve is at that height twice in one turn. It is there once on the way up, after 30°, and once on the way down, after 150°.

In symbols

sin x = 0.5 gives x = 30° or x = 150°.

Worked example

Solve cos x = 0.5 for x between 0° and 360°.

  1. The line y = 0.5 meets the cos curve twice between 0° and 360°.
  2. The calculator gives the first angle: cos⁻¹ 0.5 = 60°.
  3. The cos wave is symmetric about 180°. From 60° to 180° is 180° − 60° = 120°.
  4. The second angle is 120° on the other side of 180°: 180° + 120° = 300°.

Answer: x = 60° or x = 300°.

Try it

Measure the sine on a circle

You need: paper, a ruler, a protractor, a pencil.

  1. Draw a circle with radius 10 cm. Draw a horizontal line through its centre.
  2. With the protractor, draw radius lines at 30°, 60° and 90° above the horizontal line.
  3. Measure the height of the end of each radius above the horizontal line. Divide each height by 10.

What you will see: The heights are about 5 cm, 8.7 cm and 10 cm. Divided by 10 they give 0.5, 0.87 and 1. These are sin 30°, sin 60° and sin 90°.

If you cannot try it, this is what happens.
A table with the heads angle, height (cm) and ÷ 10. Rows: 30° and 5 and 0.5; 60° and 8.7 and 0.87; 90° and 10 and 1.angleheight (cm)÷ 1030°50.560°8.70.8790°101

A wrong idea many people have

sin x = 0.5 has only one solution, 30°.

Why it feels right: The calculator gives one answer when you press sin⁻¹, so it seems that there is only one.

The test that breaks it: The wheel point is at height 0.5 at 30°. It is at the same height again at 150°. So sin 150° = 0.5 as well.

What is true: Between 0° and 360° the line y = 0.5 meets the sin wave twice. The calculator gives the first angle. Use symmetry to find the second.

A wrong idea many people have

tan x stays between −1 and 1, like sin x.

Why it feels right: Sin, cos and tan are written alike and come from the same triangle.

The test that breaks it: tan 45° = 1, but tan 80° is about 5.7. The values keep growing as the angle gets close to 90°.

What is true: Only sin x and cos x stay between −1 and 1. The tan graph climbs without limit towards the asymptotes at 90° and 270°.

Key words

period
the length of one full wave before the graph repeats. For sin and cos it is 360°.
sine wave
the wave shaped graph of y = sin x.

Back to your guess

The answer is B: A wave that rises to 1, falls to −1, and comes back to 0.

The graph of y = sin x is a smooth wave. It starts at 0 and rises to 1 at 90°. It falls through 0 at 180° to −1 at 270°, and is back at 0 at 360°. The wave never goes above 1 or below −1, and it repeats every 360°.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The graph shows y = cos x, and a dot at 120°. What is cos 120°?

    A graph with the x axis x° and the y axis y. A wave for y = cos x starts at 1 and falls. A dot on the wave at 120 degrees has dashed lines to both axes.180360−1−0.50.510x°y
    Tap your answer.

    1. Not this one. The value −1 is where the wave is lowest, at 180°. At 120° it is not as low. Go to M4.10.5. Try another.

    2. Not this one. The size is right but the sign is wrong. At 120° the dot is below the x axis, so the value is negative. Go to M4.10.5. Try another.

    3. Yes, that is right. Follow the dashed line from the dot across to the y axis: it reaches −0.5. The dot is below the axis, so cos 120° is negative.

    See the answer

    −0.5

    Follow the dashed line from the dot across to the y axis: it reaches −0.5. The dot is below the axis, so cos 120° is negative.

    • If you chose A: The value −1 is where the wave is lowest, at 180°. At 120° it is not as low. Go to M4.10.5.
    • If you chose B: The size is right but the sign is wrong. At 120° the dot is below the x axis, so the value is negative. Go to M4.10.5.

  2. A calculator gives sin⁻¹ 0.9 = 64°, to the nearest degree. Which are the two angles between 0° and 360° with sin x = 0.9?

    Tap your answer.

    1. Not this one. A calculator shows only the first angle. The line y = 0.9 meets the wave twice, once on the way up and once on the way down. Go to M4.10.5. Try another.

    2. Not this one. That uses the symmetry of the cos wave about 180°. The sin wave is symmetric about 90°. Go to M4.10.5. Try another.

    3. Not this one. The angle 180° + 64° is below the axis, where sin x is negative. Go to M4.10.5. Try another.

    4. Yes, that is right. The calculator gives 64°. The sin wave is symmetric about 90°, so the other angle is 180° − 64° = 116°. Check: both angles are 26° from 90°.

    See the answer

    64° and 116°

    The calculator gives 64°. The sin wave is symmetric about 90°, so the other angle is 180° − 64° = 116°. Check: both angles are 26° from 90°.

    • If you chose A: A calculator shows only the first angle. The line y = 0.9 meets the wave twice, once on the way up and once on the way down. Go to M4.10.5.
    • If you chose B: That uses the symmetry of the cos wave about 180°. The sin wave is symmetric about 90°. Go to M4.10.5.
    • If you chose C: The angle 180° + 64° is below the axis, where sin x is negative. Go to M4.10.5.

  3. Which of these values can sin x never take?

    Tap your answer.

    1. Not this one. Sin x reaches −1 at 270°, so it can take this value. Go to M4.10.5. Try another.

    2. Not this one. Sin x is 0 at 0°, 180° and 360°, so it can take this value. Go to M4.10.5. Try another.

    3. Not this one. This value lies between −1 and 1, so sin x takes it at some angle. Go to M4.10.5. Try another.

    4. Yes, that is right. The sin wave stays between −1 and 1. The value 1.5 is above the top of the wave, so no angle has sin x = 1.5.

    See the answer

    1.5

    The sin wave stays between −1 and 1. The value 1.5 is above the top of the wave, so no angle has sin x = 1.5.

    • If you chose A: Sin x reaches −1 at 270°, so it can take this value. Go to M4.10.5.
    • If you chose B: Sin x is 0 at 0°, 180° and 360°, so it can take this value. Go to M4.10.5.
    • If you chose C: This value lies between −1 and 1, so sin x takes it at some angle. Go to M4.10.5.

  4. The graph of y = tan x has an asymptote at one of these angles. Which?

    Tap your answer.

    1. Yes, that is right. Tan x is sin x divided by cos x. At 90° cos x is 0, so the division fails and the graph has an asymptote.

    2. Not this one. At 45° the graph is at 1, a normal point. The asymptotes come where cos x is 0. Go to M4.10.5. Try another.

    3. Not this one. At 180° tan x is 0, because sin 180° is 0 and cos 180° is not. The graph crosses the x axis there. Go to M4.10.5. Try another.

    See the answer

    90°

    Tan x is sin x divided by cos x. At 90° cos x is 0, so the division fails and the graph has an asymptote.

    • If you chose B: At 45° the graph is at 1, a normal point. The asymptotes come where cos x is 0. Go to M4.10.5.
    • If you chose C: At 180° tan x is 0, because sin 180° is 0 and cos 180° is not. The graph crosses the x axis there. Go to M4.10.5.

  5. Solve cos x = −0.5 for x between 0° and 360°.

    Tap your answer.

    1. Not this one. These are the angles for cos x = 0.5. For −0.5 the wave must be below the axis. Go to M4.10.5. Try another.

    2. Not this one. These two are 30° either side of 180°, where cos x is about −0.87. The value −0.5 comes at 120° and 240°. Go to M4.10.5. Try another.

    3. Not this one. The cos wave is symmetric about 180°, so the two angles are the same distance either side of 180°. 120° is 60° before 180°, so the other is 60° after it. Go to M4.10.5. Try another.

    4. Yes, that is right. The calculator gives cos⁻¹(−0.5) = 120°. The cos wave is symmetric about 180°, so the other angle is 360° − 120° = 240°. Check: 120° and 240° are both 60° from 180°.

    See the answer

    x = 120° or x = 240°

    The calculator gives cos⁻¹(−0.5) = 120°. The cos wave is symmetric about 180°, so the other angle is 360° − 120° = 240°. Check: 120° and 240° are both 60° from 180°.

    • If you chose A: These are the angles for cos x = 0.5. For −0.5 the wave must be below the axis. Go to M4.10.5.
    • If you chose B: These two are 30° either side of 180°, where cos x is about −0.87. The value −0.5 comes at 120° and 240°. Go to M4.10.5.
    • If you chose C: The cos wave is symmetric about 180°, so the two angles are the same distance either side of 180°. 120° is 60° before 180°, so the other is 60° after it. Go to M4.10.5.

  6. From an earlier chapter

    Which of these is a growth curve?

    Tap your answer.

    1. Not this one. The base 0.7 is between 0 and 1, so each step multiplies by 0.7. The values fall, so this is a decay curve. Go to M4.10.4. Try another.

    2. Not this one. Any power of 1 is 1, so this graph is a flat line at height 1. It neither grows nor decays. Go to M4.10.4. Try another.

    3. Yes, that is right. A growth curve has a base above 1. Each step multiplies by 6, so the values rise faster and faster.

    See the answer

    y = 6ˣ

    A growth curve has a base above 1. Each step multiplies by 6, so the values rise faster and faster.

    • If you chose A: The base 0.7 is between 0 and 1, so each step multiplies by 0.7. The values fall, so this is a decay curve. Go to M4.10.4.
    • If you chose B: Any power of 1 is 1, so this graph is a flat line at height 1. It neither grows nor decays. Go to M4.10.4.

  7. sin 40° = 0.64, to 2 decimal places. Find the other angle between 0° and 360° with the same sine.

    See the answer

    140

    The sin wave is symmetric about 90°. The other angle is 180° − 40° = 140°.

  8. cos 70° = 0.34, to 2 decimal places. Find the other angle between 0° and 360° with the same cosine.

    See the answer

    290

    The cos wave is symmetric about 180°. The other angle is 360° − 70° = 290°.

Keep it

Copy this into your exercise book.

Sin and cos are waves between −1 and 1, repeating every 360°. Sin starts at 0, cos starts at 1. To solve sin x = 0.5, find 30°, then use symmetry for 150°.

Draw: A sin wave and a dashed cos wave from 0° to 360°. Add a horizontal line at 0.5 that meets the sin wave twice.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (956 KB). It opens on any phone.