Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — finding a side with Pythagoras
A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the third side.
- 1The unknown is a shorter side, not the hypotenuse.Identify this first — it decides whether you add or subtract.
- 2a2=132−52Finding a shorter side means subtracting.
- 3=169−25=144Square each side before subtracting. (13−5)2=64 is not the same thing.
- 4a=144=12 cmSensible: 12 is less than the hypotenuse 13, as a shorter side must be.
Example 2 — finding an angle with trigonometry
In a right-angled triangle the side opposite an angle is 6 cm and the adjacent side is 8 cm. Find the angle.
- 1You have the opposite and the adjacent, so use tangent.TOA — pick the ratio containing the two sides you know.
- 2tanθ=86=0.75Opposite over adjacent.
- 3θ=tan−1(0.75)The inverse function turns a ratio back into an angle.
- 4θ=36.9° (1 d.p.)Check your calculator is in degrees: sin30° must give exactly 0.5.
Example 3 — angles in a regular polygon
Find each interior angle of a regular hexagon.
- 1The exterior angles of any polygon add to 360°.True whatever the number of sides.
- 2Each exterior angle =6360°=60°Regular means all six are equal.
- 3An interior angle and its exterior angle lie on a straight line.So they add to 180°, not 90°.
- 4Interior angle =180°−60°=120°Check against the other formula: (6−2)×180÷6=720÷6=120 ✓