Right-Angled Trigonometry and Exact Values
Trigonometry connects an angle in a right triangle with ratios of its sides. This lesson takes you from choosing a ratio to finding lengths, angles and exact answers.
Foundation & Higher. You should know right triangles, Pythagoras and basic rearranging.
Name the sides from the chosen angle
Start with an acute angle θ in a right-angled triangle. The hypotenuse is opposite 90°. The opposite side is across from θ; the adjacent side is next to θ but is not the hypotenuse. Opposite and adjacent swap if you choose the other acute angle.
Choose a ratio
Similar right triangles with the same acute angle have the same side ratios. These ratios are called sine, cosine and tangent:
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
SOH CAH TOA is a reminder of these three relationships. Label the two sides involved, choose the ratio containing both, substitute, and rearrange. Set your calculator to degrees.
Find a side
Worked example — multiply
A right triangle has hypotenuse 10 cm and angle θ=35°. Find the opposite side x. Use sine: sin35°=x/10, so x=10sin35°≈5.74 cm (3 significant figures).
Worked example — the unknown is underneath
The side adjacent to 40° is 7 cm; find hypotenuse h. cos40°=7/h. Multiply by h, then divide by cos40°: h=7/cos40°≈9.14 cm. Multiplying 7 by cos40° would wrongly give a hypotenuse shorter than 7.
Worked example — tangent
A point is 12 m horizontally from the foot of a vertical tower. Its angle of elevation to the top is 28°. If the observation is at ground level, height=12tan28°≈6.38 m. If a person measures from eye level, add the stated eye height to this vertical difference.
Find an angle
An inverse trig function recovers an angle from a ratio. The calculator key sin⁻¹ means inverse sine, not 1/sin. Use cos⁻¹ or tan⁻¹ in the same way.
Worked example
Opposite=5 cm and adjacent=12 cm. tanθ=5/12, so θ=tan⁻¹(5/12)≈22.6° (1 decimal place). Both lengths must use the same units.
Exact values without a calculator
Learn these values. A dash means tan90° is undefined, because its denominator cos90° is zero.
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | — |
A right isosceles triangle with legs 1,1 has hypotenuse √2, giving the 45° values. Bisect an equilateral triangle of side 2: the right triangle has hypotenuse 2, short leg 1 and long leg √3. That gives the 30° and 60° values. Also 1/√3=√3/3.
Worked example — keep the answer exact
With hypotenuse 8 cm and angle 60°, opposite=8sin60°=8×√3/2=4√3 cm. Do not replace √3 with a rounded decimal in an exact-answer question.
Bearings and multi-step questions
Draw a north line at the start of a bearing. For a 10 km journey on bearing 060°, the right triangle has northward side adjacent to 60° and eastward side opposite it. Northward distance=10cos60°=5 km; eastward distance=10sin60°=5√3 km. The journey itself is the hypotenuse. For a longer route, identify each right triangle and retain unrounded intermediate results.
Think Like a Mathematician
Before using a calculator, compare the unknown with the hypotenuse. For an acute angle, sine and cosine are between 0 and 1, so a shorter side found by multiplying the hypotenuse should be smaller.
Common Mistakes
Do not label opposite before choosing the angle. Do not use the hypotenuse as the adjacent side. Avoid radians mode. Use inverse trig for an angle and rearrange carefully when the unknown is in the denominator.
Practice — right-angled trigonometry
1. Hypotenuse 12 cm, angle 30°. Find the opposite side exactly.
12sin30°=12×½=6 cm.
2. Adjacent side 9 m, angle 60°. Find the hypotenuse exactly.
cos60°=9/h, so h=9/½=18 m.
3. Adjacent side 8 cm, angle 45°. Find the opposite side.
x=8tan45°=8 cm.
4. Opposite 7 cm, hypotenuse 10 cm. Find the angle to 1 decimal place.
θ=sin⁻¹(7/10)≈44.4°.
5. Give exact values of sin45°, cos60° and tan30°.
√2/2, ½, 1/√3 (or √3/3).
6. A 20 km journey is on bearing 030°. Find its eastward component exactly.
East is opposite 30° measured from north: 20sin30°=10 km.
7. Opposite side 6 cm, angle 25°. Find the adjacent side to 3 significant figures.
tan25°=6/a, so a=6/tan25°≈12.9 cm.
Frequently Asked Questions
Are exact values Higher-only?
No. The listed exact values are Foundation and Higher content.
When should I use Pythagoras instead?
Use Pythagoras when two side lengths of a right triangle are known and you need the third. Use trig when an acute angle and side are involved, or when you need an angle.
Continue Learning
Previous: Pythagoras and Coordinate Distances
Next: 3D Pythagoras and Trigonometry
Foundation route: continue to Vectors and Vector Arithmetic. Lessons 16–18 develop the Higher route.