Pythagoras and trigonometry

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Pythagoras' Theorem relates the lengths of the sides of a right-angled triangle.

Trigonometry is the manipulation of the lengths and angles of all triangles. 

Key Concepts

Pythagoras' theorem

The sum of the squares of the two shorter sides equals the square of the hypotenuse (longest side).

\(a^2+b^2=c^2\)

 These examples are worth committing to memory:

  • A triangle of short sides 3 and 4 will have a long side length 5.
  • An isosceles triangle of short sides 1 and 1 will have a long side length \(\sqrt2 \) (incidentally an irrational number!).

Pythagoras' Theorem also applies in 3-dimensions.

We note that triangle ABD has a right angle at B (since AB is vertical and AD is horizontal). To find the length of side BD we use Pythagoras: \(BD^2=BC^2 +CD^2\)

Length AD can then be obtained using substitution: \(AD^2=AB^2+BD^2 = AB^2+BC^2+CD^2 \)

 

Trigonometry

Trigonometry is the calculation of angles and lengths of triangles using maths. The longest side of a triangle is referred to as the hypotenuse and we usually give the unknown angle the symbol 'theta' (\(\theta \)).

 In IB Physics you are only required to perform trigonometry for right angled triangles.

Calculating angles

The sides of a right angled triangle are always in the same ratio for a given angle. These ratios are called sine cosine and tangent (sin, cos and tan). To define which side is which they are given names.

\(\sin \theta\) = Opposite/Hypotenuse

\(\cos \theta\) = Adjacent/Hypotenuse

\(\tan\theta\) = Opposite/Adjacent

We can remember these equations using SOH CAH TOA. Look again at the equations above to see where this mnemonic comes from.

Always remember to check the mode of your calculator when using sin, cos and tan. An angle of between 0 and 1 is probably in radians.

Trigonometric identities

A trigonometric identity is an equation that applies for any value of angle. The identity that you must learn is:

 \(\tan \theta={\sin \theta\over \cos\theta}\)

This relationship might also come in useful for understanding certain derivations, but needn't be committed to memory:

\(\sin^2\theta+\cos^2\theta=1\)

 

Calculating lengths

By moving the blue point try changing the lengths of the triangle in the simulation below. Note that the ratio sin, cos and tan are constant for a given angle.

If you know the angle and the length of the hypotenuse then you can calculate the adjacent:

Adjacent = Hypotenuse x \(\cos\theta\)

A calculator can be used to find the cosine. Use the calculator below to find cos (60°) and show that it agrees with the triangle above.


Web 2.0 scientific calculator

 

 

 

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Use quizzes to practise application of theory.


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