Trigonometry 2.pptx

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Trigonometry 2

Angles and trigonometry

Lesson 7 of 7

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Last Lesson and Before

Angles and trigonometry

Lesson 7 of 7

Answer each one, then check.

1

Last lesson: what does SOH CAH TOA stand for?

2

Last lesson: tan 45°= x/6. Find x.

3

Round 33.749 to 1 decimal place.

4

Simplify 6/12.

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Last Lesson and Before - Answers

Angles and trigonometry

Lesson 7 of 7

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

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Learning Objectives

Angles and trigonometry

Lesson 7 of 7

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.

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Finding an Angle

Angles and trigonometry

Lesson 7 of 7

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.

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Using Inverse Sine

Angles and trigonometry

Lesson 7 of 7

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°

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Using Inverse Tangent

Angles and trigonometry

Lesson 7 of 7

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°

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PART TWO

Exact Values

Five angles whose trig values you must know without a calculator.

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Where the Exact Values Come From

Angles and trigonometry

Lesson 7 of 7

Half an equilateral triangle gives 30° and 60°; half a square gives 45°.

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.

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The Exact Values

Angles and trigonometry

Lesson 7 of 7

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

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Exact Values Without a Calculator

Angles and trigonometry

Lesson 7 of 7

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

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PART THREE

Elevation and Depression

Angles measured up or down from the horizontal.

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Angles of Elevation and Depression

Angles and trigonometry

Lesson 7 of 7

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.

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The Height of a Tree

Angles and trigonometry

Lesson 7 of 7

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

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Angles and trigonometry

Lesson 7 of 7

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

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Key Terms

Angles and trigonometry

Lesson 7 of 7

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.

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Angles and trigonometry

Lesson 7 of 7

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.

WHAT 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.

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Can I...?

Angles and trigonometry

Lesson 7 of 7

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.

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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°. 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.

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