Labelling the triangle
- Trigonometry links the sides and angles of right-angled triangles.
- The hypotenuse is the longest side, opposite the right angle. The opposite side is opposite the angle you are using. The adjacent side is next to that angle (but is not the hypotenuse).
SOHCAHTOA
- sin θ = opposite ÷ hypotenuse. cos θ = adjacent ÷ hypotenuse. tan θ = opposite ÷ adjacent.
- Finding a side: if the hypotenuse is 10 cm and the angle is 30°, the opposite side is 10 × sin 30° = 5 cm.
- Finding an angle: use the inverse button. If the opposite side is 5 and the adjacent side is 12, θ = tan⁻¹(5 ÷ 12) = 22.6°.
Exact values
- sin 0° = 0, sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1.
- cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½, cos 90° = 0.
- tan 0° = 0, tan 30° = √3/3 (= 1/√3), tan 45° = 1, tan 60° = √3.
Elevation, depression and 3D
- The angle of elevation is measured up from the horizontal. The angle of depression is measured down from the horizontal.
- (Higher) In 3D problems, find a right-angled triangle inside the shape, then use Pythagoras or trigonometry.
Key terms
- Hypotenuse
- The longest side of a right-angled triangle, opposite the right angle.
- Opposite
- The side opposite the angle you are using.
- Adjacent
- The side next to the angle you are using, which is not the hypotenuse.
- Sine (sin)
- Opposite ÷ hypotenuse.
- Cosine (cos)
- Adjacent ÷ hypotenuse.
- Tangent (tan)
- Opposite ÷ adjacent.
- Angle of elevation
- The angle measured upwards from the horizontal.
- Angle of depression
- The angle measured downwards from the horizontal.