Rukasha Maths

GCSE maths Beta

Pythagoras and right-angled trigonometry · Revision guide

Foundation · Geometry · 2 pages in the PDF

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Learning objectives

  • Calculate a missing side in a right-angled triangle.
  • Choose a trigonometric ratio for a side or angle.

Key ideas

The hypotenuse is opposite the right angle. Pythagoras applies only to right-angled triangles.

a2+b2=c2(hypotenuse c)a^2+b^2=c^2\quad\text{(hypotenuse }c\text{)}

Add squares to find the hypotenuse; subtract squares to find a shorter side. Then take a square root.

Label opposite and adjacent relative to the chosen acute angle.

sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\qquad\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}

Use the tangent ratio when the hypotenuse is not involved.

tan⁡θ=oppositeadjacent\tan\theta=\frac{\text{opposite}}{\text{adjacent}}

For a missing angle, use the appropriate inverse ratio in degree mode.

Worked examples

Example 1

Find the unknown shorter side of a right-angled triangle.

c=13 cm,a=5 cm:x2+52=132c=13\text{ cm},\quad a=5\text{ cm}:\qquad x^2+5^2=13^2
x2=169−25=144x^2=169-25=144

A length takes the positive root.

x=144=12 cmx=\sqrt{144}=12\text{ cm}
12 cm12\text{ cm}

Example 2

Find the side opposite the acute angle in a right-angled triangle.

θ=30∘,c=10 cm:sin⁡30∘=o10\theta=30^\circ,\quad c=10\text{ cm}:\qquad\sin30^\circ=\frac{o}{10}
o=10sin⁡30∘=10×12=5 cmo=10\sin30^\circ=10\times\frac12=5\text{ cm}
5 cm5\text{ cm}

Watch out

A squared length is not the final length. Take the square root and check that the hypotenuse is the longest side.