EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Trigonometry 2
Angles and trigonometry · Lesson 7 of 7
Last Lesson and Before
Answer each one, then check.
1. Last lesson: what does SOH CAH TOA stand for?
sin = opp/hyp, cos = adj/hyp, tan = opp/adj
2. Last lesson: tan 45°= x/6. Find x.
6
3. Round 33.749 to 1 decimal place.
33.7
4. Simplify 6/12.
½
Learning Objectives
1. Use inverse trig functions to find a missing angle.
2. Know the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°.
3. Solve 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 θ= 5/9.
▸ Inverse. θ= sin ⁻¹(5/9).
▸ Round. Angles are usually given to 1 decimal place.
Using Inverse Sine
|
A right-angled triangle has hypotenuse 9 cm. The side opposite angle θ is 5 cm. Work out θ to 1 decimal place. |
1. Known: O = 5 and H = 9, so sine
sin θ= 5/9
2. Inverse sine
θ= sin ⁻¹(5/9)
3. Calculate
θ= 33.748…
Answer: 33.7°
Using Inverse Tangent
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A right-angled triangle has sides 8 cm (opposite θ) and 11 cm (adjacent to θ). Work out θ to 1 decimal place. |
1. O and A, so tangent
tan θ= 8/11
2. Inverse tangent
θ= tan ⁻¹(8/11)
3. Calculate
θ= 36.027…
Answer: 36.0°
PART TWO
Exact Values
Five angles whose trig values you must know without a calculator.
Where the Exact Values Come From
|
Half of an equilateral triangle of side 2 has sides 1, 2 and √3 (by Pythagoras), so sin 30°= ½ and tan 60°= √3. A right-angled isosceles triangle with shorter sides 1 has hypotenuse √2, so tan 45°= 1. |
Half an equilateral triangle gives 30° and 60°; half a square gives 45°. |
The Exact Values
|
Angle |
sin |
cos |
tan |
|---|---|---|---|
|
0° |
0 |
1 |
0 |
|
30° |
½ |
√3/2 |
1/(√3) = √3/3 |
|
45° |
1/(√2) = √2/2 |
√2/2 |
1 |
|
60° |
√3/2 |
½ |
√3 |
|
90° |
1 |
0 |
not defined |
Exact Values Without a Calculator
|
A right-angled triangle has hypotenuse 10 cm and an angle of 30°. Work out the side opposite the 30° angle. Do not use a calculator. |
1. O and H, so sine
x = 10sin 30°
2. Exact value
sin 30°= ½
3. Calculate
x = 10 × ½ = 5
Answer: 5 cm
PART THREE
Elevation and Depression
Angles measured up or down from the horizontal.
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°. Work out the height of the tree, to 1 decimal place. |
1. Sketch: A = 20 (along the ground), O = height
tan 32°= h/20
2. Multiply by 20
h = 20 tan 32°
3. Calculate
h = 12.497…
Answer: 12.5 m
Case Study
|
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 |
Key Terms
|
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. |
|
Your Task: Measure the School
20 minutes
|
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 θ+ eye height. For example, 20 m away, 32°, eye height 1.5 m: 20tan 32°+ 1.5 = 14.0 m.
Can I...?
☐ Find an angle using sin ⁻¹, cos ⁻¹ or tan ⁻¹.
☐ Round an angle to 1 decimal place.
☐ Recall the exact values for 0°, 30°, 45°, 60° and 90°.
☐ Use exact values without a calculator.
☐ Draw a diagram for an elevation or depression problem.
☐ Solve elevation and depression problems.
Summary
✓ 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.
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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°. How far is the boat from the foot of the cliff? (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. |