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Maths · Angles and trigonometry

Trigonometry 1

Sine, cosine and tangent link the angles of a right-angled triangle to its sides. Label the sides, choose the right ratio, and you can find any missing side.

  • 6 key terms
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Last Lesson and Before

Answer each one, then check.

  1. 1

    Last lesson: how far apart are \((0, 0)\) and \((6, 8)\)?

    Show answerHide answer

    10

  2. 2

    Solve \(\frac{x}{4} = 3\).

    Show answerHide answer

    \(x = 12\)

  3. 3

    Solve \(\frac{12}{x} = 3\).

    Show answerHide answer

    \(x = 4\)

  4. 4

    Round 6.8829 to 3 significant figures.

    Show answerHide answer

    6.88

Learning Objectives

  1. 1Label the hypotenuse, opposite and adjacent sides from a given angle.
  2. 2Know the sine, cosine and tangent ratios.
  3. 3Choose the right ratio for a problem.
  4. 4Find a missing side of a right-angled triangle.

SOH CAH TOA

\(\theta\) is the Greek letter theta, often used for an angle.

  • Sine

    Formula: \(\sin\theta = \dfrac{\text{opp}}{\text{hyp}}\). Uses: Opposite and hypotenuse

  • Cosine

    Formula: \(\cos\theta = \dfrac{\text{adj}}{\text{hyp}}\). Uses: Adjacent and hypotenuse

  • Tangent

    Formula: \(\tan\theta = \dfrac{\text{opp}}{\text{adj}}\). Uses: Opposite and adjacent

Finding a Missing Side

The same five steps every time.

  1. 1 Label

    Label the sides O, A and H from the given angle.
  2. 2 Choose

    Tick the side you know and the side you want. The two letters pick the ratio: O and H - sine; A and H - cosine; O and A - tangent.
  3. 3 Write

    Write the ratio with the numbers in, e.g. \(\sin 35^\circ = \dfrac{x}{12}\).
  4. 4 Rearrange

    If \(x\) is on top, multiply. If \(x\) is on the bottom, swap \(x\) and the trig value.
  5. 5 Calculate

    Check your calculator is in degrees (D on the screen), then round.

The Unknown on Top

A right-angled triangle has hypotenuse 15 cm. Angle \(\theta = 42^\circ\). Work out the side opposite \(\theta\), to 1 decimal place.

Show the solutionHide the solution
  1. 1 Known: H = 15. Wanted: O. O and H means sine \(\sin 42^\circ = \dfrac{x}{15}\)
  2. 2 Multiply both sides by 15 \(x = 15 \sin 42^\circ\)
  3. 3 Calculate \(x = 10.036\ldots\)

Answer10.0 cm

Using Tangent

A right-angled triangle has an angle of \(55^\circ\). The side adjacent to it is 8 cm. Work out the side opposite the \(55^\circ\) angle, to 1 decimal place.

Show the solutionHide the solution
  1. 1 Known: A = 8. Wanted: O. O and A means tangent \(\tan 55^\circ = \dfrac{x}{8}\)
  2. 2 Multiply by 8 \(x = 8 \tan 55^\circ\)
  3. 3 Calculate \(x = 11.425\ldots\)

Answer11.4 cm

The Unknown on the Bottom

A right-angled triangle has an angle of \(23^\circ\). The side opposite it is 6 cm. Work out the hypotenuse, to 1 decimal place.

Show the solutionHide the solution
  1. 1 Known: O = 6. Wanted: H. O and H means sine \(\sin 23^\circ = \dfrac{6}{x}\)
  2. 2 \(x\) is on the bottom, so swap \(x\) and \(\sin 23^\circ\) \(x = \dfrac{6}{\sin 23^\circ}\)
  3. 3 Calculate \(x = 15.355\ldots\)
  4. 4 Check: the hypotenuse is the longest side \(15.4 > 6\)

Answer15.4 cm

Trig Relay

In pairs. For each triangle, one partner labels the sides and chooses the ratio; the other writes the equation and calculates. Swap roles each time. (a) H = 10, angle \(50^\circ\), find A. (b) A = 14, angle \(36^\circ\), find O. (c) A = 25, angle \(20^\circ\), find H. (d) H = 3, angle \(12^\circ\), find O.

1. Label O, A, H.

2. Choose the ratio.

3. Rearrange and calculate.

A good answer shows: (a) \(10\cos 50^\circ = 6.43\) (b) \(14\tan 36^\circ = 10.17\) (c) \(\dfrac{25}{\cos 20^\circ} = 26.60\) (d) \(3\sin 12^\circ = 0.62\), all to 2 decimal places.

Can I...?

  1. 1Label the hypotenuse, opposite and adjacent sides.
  2. 2Recall SOH CAH TOA.
  3. 3Choose the right ratio.
  4. 4Find a side when it is on the top of the fraction.
  5. 5Find a side when it is on the bottom of the fraction.
  6. 6Check my calculator is in degrees.

Summary & Exam Focus

  • Label from the angle: hypotenuse, opposite, adjacent.
  • SOH CAH TOA picks the ratio.
  • \(x\) on top: multiply. \(x\) on the bottom: divide.
  • Degree mode, then round at the end.

Exam focus

Triangle ABC is right-angled at B. AB = 7 cm and angle ACB = \(40^\circ\). Work out the length of BC. Give your answer to 3 significant figures. (3 marks) (3 marks)

Write the ratio with the numbers in before you rearrange - it is usually worth a method mark on its own, even if the calculator step goes wrong.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Hypotenuse
The side opposite the right angle; the longest side.
Opposite
The side across from the angle being used.
Adjacent
The side next to the angle being used, which is not the hypotenuse.
Sine (sin)
\(\sin\theta = \frac{\text{opp}}{\text{hyp}}\).
Cosine (cos)
\(\cos\theta = \frac{\text{adj}}{\text{hyp}}\).
Tangent (tan)
\(\tan\theta = \frac{\text{opp}}{\text{adj}}\).

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Calculator 2 marks

    A right-angled triangle has hypotenuse 12 cm. One of its angles is \(35^\circ\). Work out the length of the side opposite the \(35^\circ\) angle. Give your answer to 3 significant figures.

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    Model answer

    \(x = 12 \sin 35^\circ = 6.882\ldots = 6.88\) cm

    Mark scheme

    • \(\sin 35^\circ = \dfrac{x}{12}\) — M1
    • 6.88 — A1
  2. Question 2 Calculator 3 marks

    Triangle ABC is right-angled at B. AB = 7 cm and angle ACB = \(40^\circ\). Work out the length of BC. Give your answer to 3 significant figures.

    A right-angled triangle ABC with the right angle at B, AB 7 cm and angle C 40 degrees.
    Show answerHide answer

    Model answer

    From angle C, AB is opposite and BC is adjacent. \(\tan 40^\circ = \dfrac{7}{BC}\), so \(BC = \dfrac{7}{\tan 40^\circ} = 8.342\ldots = 8.34\) cm

    Mark scheme

    • \(\tan 40^\circ = \dfrac{7}{BC}\) — M1
    • \(BC = \dfrac{7}{\tan 40^\circ}\) — M1
    • 8.34 — A1
  3. Question 3 Calculator 3 marks

    A right-angled triangle has an angle of \(28^\circ\). The side adjacent to this angle is 9 cm. Work out the length of the hypotenuse. Give your answer to 1 decimal place.

    Show answerHide answer

    Model answer

    \(\cos 28^\circ = \dfrac{9}{h}\), so \(h = \dfrac{9}{\cos 28^\circ} = 10.193\ldots = 10.2\) cm

    Mark scheme

    • \(\cos 28^\circ = \dfrac{9}{h}\) — M1
    • \(h = \dfrac{9}{\cos 28^\circ}\) — M1
    • 10.2 — A1
  4. Question 4 Calculator 3 marks

    Wheelchair ramps must not be too steep. The rules say a ramp must rise no more than 1 m for every 12 m along the ground. A ramp rises at an angle of \(5^\circ\) along 14 m of horizontal ground. Does it meet the rule? Show your working.

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    Model answer

    Rise \(= 14 \tan 5^\circ = 1.224\ldots\) m. The rule allows \(14 \div 12 = 1.166\ldots\) m. \(1.22 > 1.17\), so the ramp does not meet the rule.

    Mark scheme

    • \(14 \tan 5^\circ\) — M1
    • 1.22 and 1.17 (or an equivalent comparison, e.g. \(\tan 5^\circ = 0.087 > \frac{1}{12} = 0.083\)) — M1
    • No, with correct figures — C1

Quick check

  1. In a right-angled triangle you know the adjacent side and want the opposite side. Which ratio do you use?

    1. ASine
    2. BCosine
    3. CTangent
    4. DPythagoras
    Show answerHide answer

    C: Tangent

    Opposite and adjacent: TOA, so tangent.

  2. \(\cos 60^\circ = \dfrac{x}{10}\). What is \(x\)?

    1. A20
    2. B0.05
    3. C8.66
    4. D5
    Show answerHide answer

    D: 5

    \(x = 10\cos 60^\circ = 10 \times 0.5 = 5\).

  3. \(\sin 30^\circ = \dfrac{4}{x}\). What is \(x\)?

    1. A2
    2. B8
    3. C4.5
    4. D0.125
    Show answerHide answer

    B: 8

    \(x = \dfrac{4}{\sin 30^\circ} = \dfrac{4}{0.5} = 8\).

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