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Maths · Angles and trigonometry

Trigonometry 2

Running trigonometry backwards to find a missing angle, the exact trig values you must know without a calculator, and angles of elevation and depression.

  • 5 key terms
  • All boards
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Last Lesson and Before

Answer each one, then check.

  1. 1

    Last lesson: what does SOH CAH TOA stand for?

    Show answerHide answer

    \(\sin = \frac{\text{opp}}{\text{hyp}}\), \(\cos = \frac{\text{adj}}{\text{hyp}}\), \(\tan = \frac{\text{opp}}{\text{adj}}\)

  2. 2

    Last lesson: \(\tan 45^\circ = \frac{x}{6}\). Find \(x\).

    Show answerHide answer

    6

  3. 3

    Round 33.749 to 1 decimal place.

    Show answerHide answer

    33.7

  4. 4

    Simplify \(\frac{6}{12}\).

    Show answerHide answer

    \(\frac{1}{2}\)

Learning Objectives

  1. 1Use inverse trig functions to find a missing angle.
  2. 2Know the exact values of sin, cos and tan for \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\) and \(90^\circ\).
  3. 3Solve problems with angles of elevation and depression.

Finding an Angle

To find an angle, use the inverse function: \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\) (SHIFT then sin, cos or tan).

  • Label and choose

    Exactly as before: the two sides you know pick the ratio.

  • Write the ratio as a fraction

    \(\sin\theta = \dfrac{5}{9}\).

  • Inverse

    \(\theta = \sin^{-1}\left(\dfrac{5}{9}\right)\).

  • Round

    Angles are usually given to 1 decimal place.

Using Inverse Sine

A right-angled triangle has hypotenuse 9 cm. The side opposite angle \(\theta\) is 5 cm. Work out \(\theta\) to 1 decimal place.

Show the solutionHide the solution
  1. 1 Known: O = 5 and H = 9, so sine \(\sin\theta = \dfrac{5}{9}\)
  2. 2 Inverse sine \(\theta = \sin^{-1}\left(\dfrac{5}{9}\right)\)
  3. 3 Calculate \(\theta = 33.748\ldots\)

Answer\(33.7^\circ\)

Using Inverse Tangent

A right-angled triangle has sides 8 cm (opposite \(\theta\)) and 11 cm (adjacent to \(\theta\)). Work out \(\theta\) to 1 decimal place.

Show the solutionHide the solution
  1. 1 O and A, so tangent \(\tan\theta = \dfrac{8}{11}\)
  2. 2 Inverse tangent \(\theta = \tan^{-1}\left(\dfrac{8}{11}\right)\)
  3. 3 Calculate \(\theta = 36.027\ldots\)

Answer\(36.0^\circ\)

The Exact Values

  • \(0^\circ\)

    sin: 0. cos: 1. tan: 0

  • \(30^\circ\)

    sin: \(\frac{1}{2}\). cos: \(\frac{\sqrt{3}}{2}\). tan: \(\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)

  • \(45^\circ\)

    sin: \(\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\). cos: \(\frac{\sqrt{2}}{2}\). tan: 1

  • \(60^\circ\)

    sin: \(\frac{\sqrt{3}}{2}\). cos: \(\frac{1}{2}\). tan: \(\sqrt{3}\)

  • \(90^\circ\)

    sin: 1. cos: 0. tan: not defined

Exact Values Without a Calculator

A right-angled triangle has hypotenuse 10 cm and an angle of \(30^\circ\). Work out the side opposite the \(30^\circ\) angle. Do not use a calculator.

Show the solutionHide the solution
  1. 1 O and H, so sine \(x = 10\sin 30^\circ\)
  2. 2 Exact value \(\sin 30^\circ = \frac{1}{2}\)
  3. 3 Calculate \(x = 10 \times \frac{1}{2} = 5\)

Answer5 cm

Angles of Elevation and Depression

Both are always measured from the horizontal.

  • Angle of elevation

    The angle you look UP through from the horizontal - from the ground to the top of a tree.

  • Angle of depression

    The angle you look DOWN through from the horizontal - from a cliff top to a boat.

  • They are equal

    The angle of depression from A to B equals the angle of elevation from B to A: they are alternate angles.

  • Draw it

    Sketch the horizontal line, the right angle and the angle before you calculate.

The Height of a Tree

Jess stands 20 m from the foot of a tree on flat ground. The angle of elevation of the top of the tree from the ground where she stands is \(32^\circ\). Work out the height of the tree, to 1 decimal place.

Show the solutionHide the solution
  1. 1 Sketch: A = 20 (along the ground), O = height \(\tan 32^\circ = \dfrac{h}{20}\)
  2. 2 Multiply by 20 \(h = 20 \tan 32^\circ\)
  3. 3 Calculate \(h = 12.497\ldots\)

Answer12.5 m

Case study

Measuring Everest

In the 1800s the Great Trigonometrical Survey of India measured the whole subcontinent with triangles. From observation stations more than 160 km away, surveyors measured the angle of elevation of a remote Himalayan summit called Peak XV. In 1852 the mathematician Radhanath Sikdar worked through the calculations and found it was the highest mountain in the world. It was announced in 1856 as 29 002 feet (about 8840 m) and later named Everest. The modern figure, agreed by China and Nepal in 2020, is 8848.86 m - the 1850s trigonometry was out by less than 0.1%.

1852 Sikdar's calculations show Peak XV is the highest
8848.86 m The modern height of Everest

Measure the School

Make a simple clinometer from a protractor, a straw and a weight on a string. Stand a measured distance from a tall building or tree, measure the angle of elevation of the top, and work out its height. Remember to add your eye height.

1. Measure the distance to the base.

2. Measure the angle of elevation.

3. Calculate, then add your eye height.

A good answer shows: Height \(= d \tan\theta +\) eye height. For example, 20 m away, \(32^\circ\), eye height 1.5 m: \(20\tan 32^\circ + 1.5 = 14.0\) m.

Can I...?

  1. 1Find an angle using \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\).
  2. 2Round an angle to 1 decimal place.
  3. 3Recall the exact values for \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\) and \(90^\circ\).
  4. 4Use exact values without a calculator.
  5. 5Draw a diagram for an elevation or depression problem.
  6. 6Solve elevation and depression problems.

Summary & Exam Focus

  • Finding an angle: write the ratio, then use the inverse function.
  • Learn the exact values table - they appear on the non-calculator paper.
  • Elevation looks up and depression looks down, both from the horizontal.

Exam focus

A lighthouse stands on the edge of a cliff. The top of the lighthouse is 80 m above sea level. The angle of depression from the top of the lighthouse to a boat is \(25^\circ\). How far is the boat from the foot of the cliff? (3 marks) (3 marks)

Angles of depression are measured from the horizontal at the top - not from the vertical cliff. Use alternate angles to put the angle inside your triangle, at the boat.

Key terms

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

Inverse function
\(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\): finds the angle from a ratio.
Exact value
A trig value written exactly, as a fraction or surd, not a rounded decimal.
Angle of elevation
The angle measured up from the horizontal.
Angle of depression
The angle measured down from the horizontal.
Clinometer
An instrument for measuring angles of elevation.

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. The side adjacent to angle \(\theta\) is 7 cm. Work out the size of angle \(\theta\). Give your answer to 1 decimal place.

    Show answerHide answer

    Model answer

    \(\cos\theta = \dfrac{7}{12}\), so \(\theta = \cos^{-1}\left(\dfrac{7}{12}\right) = 54.31\ldots = 54.3^\circ\)

    Mark scheme

    • \(\cos\theta = \dfrac{7}{12}\) — M1
    • \(54.3^\circ\) — A1
  2. Question 2 Non-calculator 2 marks

    A right-angled triangle has an angle of \(60^\circ\). The hypotenuse is 14 cm. Work out the length of the side adjacent to the \(60^\circ\) angle.

    Show answerHide answer

    Model answer

    \(x = 14\cos 60^\circ = 14 \times \frac{1}{2} = 7\) cm

    Mark scheme

    • \(14\cos 60^\circ\) — M1
    • 7 cm — A1
  3. Question 3 Calculator 3 marks

    A lighthouse stands on the edge of a cliff. The top of the lighthouse is 80 m above sea level. The angle of depression from the top of the lighthouse to a boat is \(25^\circ\). Work out the distance from the boat to the foot of the cliff. Give your answer to the nearest metre.

    A lighthouse on a cliff 80 metres above the sea, with a line down to a boat and a 25 degree angle of depression marked below the horizontal at the top.
    Show answerHide answer

    Model answer

    The angle of elevation of the top from the boat is also \(25^\circ\) (alternate angles). \(\tan 25^\circ = \dfrac{80}{d}\), so \(d = \dfrac{80}{\tan 25^\circ} = 171.56\ldots = 172\) m.

    Mark scheme

    • \(25^\circ\) placed at the boat, or the angle \(65^\circ\) at T found — M1
    • \(\tan 25^\circ = \dfrac{80}{d}\), or \(80\tan 65^\circ\) — M1
    • 172 m — A1
  4. Question 4 Calculator 4 marks

    A kite string is 30 m long and is pulled tight. The string makes an angle of \(52^\circ\) with the horizontal ground. The string is held 1.2 m above the ground. (a) Work out the height of the kite above the ground, to 1 decimal place. (b) The wind drops, and the kite falls to a height of 15 m, with the same length of string, still held 1.2 m above the ground. Work out the new angle between the string and the horizontal.

    Show answerHide answer

    Model answer

    (a) \(30\sin 52^\circ = 23.64\ldots\), plus 1.2 gives 24.8 m. (b) Height above the hand \(= 15 - 1.2 = 13.8\) m. \(\sin\theta = \dfrac{13.8}{30}\), \(\theta = \sin^{-1}(0.46) = 27.4^\circ\).

    Mark scheme

    • (a) \(30\sin 52^\circ\) — M1
    • (a) 24.8 m — A1
    • (b) \(\sin\theta = \dfrac{13.8}{30}\) — M1
    • (b) \(27.4^\circ\) — A1

Quick check

  1. What is the exact value of \(\tan 45^\circ\)?

    1. A1
    2. B\(\frac{1}{2}\)
    3. C\(\frac{\sqrt{2}}{2}\)
    4. D\(\sqrt{3}\)
    Show answerHide answer

    A: 1

    A right-angled isosceles triangle has equal opposite and adjacent sides, so \(\tan 45^\circ = 1\).

  2. A right-angled triangle has opposite side 4 cm and hypotenuse 7 cm. What is the angle, to 1 decimal place?

    1. A\(29.7^\circ\)
    2. B\(55.2^\circ\)
    3. C\(34.8^\circ\)
    4. D\(0.6^\circ\)
    Show answerHide answer

    C: \(34.8^\circ\)

    \(\sin^{-1}\left(\frac{4}{7}\right) = 34.8^\circ\).

  3. The angle of elevation of the top of a tower from a point P is \(40^\circ\). What is the angle of depression of P from the top of the tower?

    1. A\(50^\circ\)
    2. B\(140^\circ\)
    3. C\(80^\circ\)
    4. D\(40^\circ\)
    Show answerHide answer

    D: \(40^\circ\)

    The horizontal at the top is parallel to the ground, so the two angles are alternate and equal.

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