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2D Trigonometry

2D Trigonometry

Pythagoras and right-angled trigonometry

Right Angled Triangles Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Pythagoras' theorem
  4. Sin, cos and tan
  5. Paper 1: trig without a calculator
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 6.1 and 6.2: Pythagoras and right-angled triangles
  • 6.1Use Pythagoras' theorem to find a missing side.
  • 6.2Use sin, cos and tan to find a side or an angle in a right-angled triangle.
  • 6.2Do it without a calculator when Paper 1 gives you the trig value.

Before you start: rearranging simple formulas. Next: Elevation, depression and bearings.

Formula bank

PythagorasLearn it\(a^{2}+b^{2}=c^{2}\)c = hypotenuse (longest side, opposite the right angle).Use it when two sides known, third side wanted.
SineLearn it\(\sin x=\dfrac{\text{opp}}{\text{hyp}}\)Use it when opposite and hypotenuse involved.
CosineLearn it\(\cos x=\dfrac{\text{adj}}{\text{hyp}}\)Use it when adjacent and hypotenuse involved.
TangentLearn it\(\tan x=\dfrac{\text{opp}}{\text{adj}}\)Use it when opposite and adjacent involved.
AngleLearn it\(x=\sin^{-1}(\ldots),\ \cos^{-1}(\ldots),\ \tan^{-1}(\ldots)\)the SHIFT key on the calculator.Use it when an angle wanted.

Pythagoras' theorem

In a right-angled triangle the square on the longest side (the hypotenuse) equals the two smaller squares added: \(6^{2}+8^{2}=100\), so the hypotenuse is 10.

General methodHow to find a missing side with Pythagoras

  1. Find the hypotenuse: the side opposite the right angle.
  2. Hypotenuse wanted? ADD the squares, then square root.
  3. Shorter side wanted? SUBTRACT: hyp² − other side², then square root.
  4. Check: the hypotenuse must be the longest side.

Example A right-angled triangle has hypotenuse 13 cm and one side 5 cm. Find the other side.

A shorter side, so subtract.

  1. \(13^{2}-5^{2}=169-25=144\)
  2. \(\sqrt{144}=12\)

Answer 12 cm

Try it yourself

1
The two shorter sides are 5 cm and 7 cm. Hypotenuse (2 d.p.)?
Show answer

Answer 8.60

2
Hypotenuse 12.5 cm, one side 7.5 cm. Other side (cm)?
Show answer

Answer 10

Sin, cos and tan

Right-angled triangle: hypotenuse, opposite and adjacent to angle xxhypotenuseoppositeadjacent
Name the sides from the angle you are using.

Label the sides from the angle x: the hypotenuse is opposite the right angle, the opposite faces x, the adjacent is next to x. Remember SOH CAH TOA.

General methodHow to answer any right-angled trig question

  1. Label hyp, opp and adj from the angle you know (or want).
  2. Tick the two sides involved: the one you know and the one you want.
  3. Pick SOH, CAH or TOA that uses exactly those two.
  4. Side on top of the fraction? Multiply (\(\text{opp}=12\sin35^\circ\)). Side on the bottom? Divide (\(\text{hyp}=9\div\cos40^\circ\)).
  5. Angle wanted? Use \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\); give it to 1 d.p.

Example Hypotenuse 12 cm, angle 35°. Find the side opposite the angle.

opp and hyp: SOH.

  1. \(\sin35^\circ=\dfrac{\text{opp}}{12}\)
  2. \(\text{opp}=12\sin35^\circ=6.88\)

Answer 6.88 cm

Example Angle 40°, adjacent side 9 cm. Find the hypotenuse.

adj and hyp: CAH. The unknown is on the bottom, so divide.

  1. \(\cos40^\circ=\dfrac{9}{\text{hyp}}\)
  2. \(\text{hyp}=\dfrac{9}{\cos40^\circ}=11.7\)

Answer 11.7 cm (3 s.f.)

Example Opposite 5 cm, adjacent 8 cm. Find the angle.

opp and adj: TOA, then inverse.

  1. \(\tan x=\dfrac58\)
  2. \(x=\tan^{-1}(0.625)=32.0^\circ\)

Answer 32.0°

Try it yourself

1
Angle 28°, adjacent 15 cm. Opposite side (2 d.p.)?
Show answer

Answer 7.98

2
Adjacent 7 cm, hypotenuse 12 cm. The angle (1 d.p.)?
Show answer

Answer 54.3

Paper 1: trig without a calculator

Paper 1 can still ask trig. It then GIVES you the value you need ("use \(\tan40^\circ=0.84\)") or uses a triangle whose sides are whole numbers.

General methodHow to do trig on the non-calculator paper

  1. Write the SOH CAH TOA ratio as normal.
  2. Put in the trig value they give you instead of pressing a key.
  3. If you know one ratio, draw the triangle: put the fraction's numbers on its sides, find the third side with Pythagoras, then read off any other ratio.

Example \(\sin x=0.6\) and x is acute. Find \(\cos x\) and \(\tan x\).

\(0.6=\frac{6}{10}\): opposite 6, hypotenuse 10.

  1. adjacent \(=\sqrt{10^{2}-6^{2}}=8\)
  2. \(\cos x=\frac{8}{10}=0.8\), \(\tan x=\frac68=0.75\)

Answer 0.8 and 0.75

Try it yourself

1
Use \(\tan40^\circ=0.84\). Angle 40°, adjacent 5 cm. Opposite side (cm)?
Show answer

Answer 4.2

2
\(\sin x=0.6\), x acute. Find \(\tan x\).
Show answer

Answer 0.75

Ways they test it

Type 1 Missing side, right angle shown

"Calculate x." (2 marks)

Spot it a right-angle box on the diagram.

  1. Two sides known and no angle? Pythagoras.
  2. An angle involved? SOH CAH TOA.

Type 2 Missing angle

"Calculate angle ABC." (2 marks)

Spot it two sides known, angle wanted.

  1. Pick the ratio.
  2. Inverse; 1 d.p.

For example Opposite 5, adjacent 8

  1. \(\tan^{-1}\frac58\)

Answer 32.0°

Type 3 Show that

"Show that AB = 6.88 cm, correct to 2 d.p." (2 marks)

Spot it 'show that' with a rounded value.

  1. Write the full working.
  2. Give MORE accuracy than asked (6.882…), then round.

Type 4 Paper 1 with a given value

"Use \(\cos50^\circ=0.64\) to find …" (2 marks)

Spot it a trig value printed in the question.

  1. Use that value in place of the calculator.

Mistakes to avoid

  • ✗ Adding squares to find a SHORTER side✓ subtractOnly the hypotenuse comes from adding.
  • ✗ Labelling from the wrong angle✓ label opp/adj from the angle you are using
  • ✗ \(9\times\cos40^\circ\) when the hyp is wanted✓ \(9\div\cos40^\circ\)The unknown is on the bottom.
  • ✗ Calculator in radians✓ check DEG modesin 30 should give 0.5.

Summary

  • Pythagoras: add for the hypotenuse, subtract for a shorter side.
  • SOH CAH TOA: choose the ratio with the two sides you are using.
  • Unknown on top: multiply. On the bottom: divide. Angle: inverse.
  • Paper 1: use the given value, or draw the triangle from the ratio.

If you see …

  • right angle + two sides, no anglePythagoras.
  • right angle + an angleSOH CAH TOA.

Practice

Practice: easy, medium, hard

Easy

1
Shorter sides 6 cm and 8 cm. Hypotenuse (cm)?
Show answer

Answer 10

2
Hypotenuse 13 cm, one side 5 cm. Other side (cm)?
Show answer

Answer 12

3
Which ratio uses opposite and adjacent? (sin, cos or tan)
Show answer

Answer tan

Medium

1
Hypotenuse 12 cm, angle 35°. Opposite side (2 d.p.)?
Guide me

What is it asking? a side

What do we know? opp + hyp: SOH

Plan \(12\sin35^\circ\)

Show working
  1. 6.88

Answer 6.88

2
Angle 40°, adjacent 9 cm. Hypotenuse (1 d.p.)?
Guide me

What is it asking? a side

What do we know? adj + hyp: CAH

Plan \(9\div\cos40^\circ\)

Show working
  1. 11.7

Answer 11.7

3
Opposite 5 cm, adjacent 8 cm. Angle (1 d.p.)?
Guide me

What is it asking? an angle

What do we know? TOA

Plan \(\tan^{-1}(5/8)\)

Show working
  1. 32.0

Answer 32.0

Hard

1
Adjacent 7 cm, hypotenuse 12 cm. Find the angle (1 d.p.).
Guide me

What is it asking? an angle

What do we know? adj and hyp: CAH

Plan \(\cos^{-1}(7/12)\)

Show working
  1. \(54.31\ldots\)

Answer 54.3

PartAnswer / workingMarksGuidance
\(\cos x=\frac{7}{12}\)M1
54.3A1
Total: 2 marks
2
\(\sin x=0.6\), x acute. Find \(\cos x\) without a calculator.
Guide me

What is it asking? a ratio

What do we know? opp 6, hyp 10

Plan adjacent by Pythagoras: 8

Show working
  1. \(\frac{8}{10}\)

Answer 0.8

PartAnswer / workingMarksGuidance
adj = 8M1
0.8A1
Total: 2 marks
3
Shorter sides 5 cm and 7 cm. Hypotenuse (2 d.p.)?
Guide me

What is it asking? a side

What do we know? add squares: 74

Plan \(\sqrt{74}\)

Show working
  1. 8.60

Answer 8.60

PartAnswer / workingMarksGuidance
\(5^2+7^2\)M1
8.60A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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