Maths
What is trigonometry?
Quick answer
Trigonometry links the angles and sides of right-angled triangles using sine, cosine and tangent. It lets you find a missing side or angle. SOH CAH TOA helps you remember the three ratios.
In one sentence
Trigonometry uses the fixed relationship between the angles and sides of right-angled triangles to find unknown lengths and angles.
Explained
If two right-angled triangles have the same angles, they have the same shape, even if one is bigger. That means the ratios of their sides stay the same. Trigonometry gives those ratios names.
To use them, label the sides of a right-angled triangle from the point of view of the angle you are looking at (call it θ, the Greek letter theta):
- Hypotenuse: the longest side, opposite the right angle.
- Opposite: the side facing the angle θ.
- Adjacent: the side next to θ that is not the hypotenuse.
The three ratios are
- sin θ = opposite ÷ hypotenuse
- cos θ = adjacent ÷ hypotenuse
- tan θ = opposite ÷ adjacent
Remember them as SOH CAH TOA.
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Finding a side. A 5 m ladder leans against a wall and makes a 60° angle with the ground. How high does it reach? The wall height is opposite the angle, and the ladder is the hypotenuse, so use sine. sin 60° = height ÷ 5, so height = 5 × sin 60° ≈ 5 × 0.866 = 4.33 m.
Finding an angle. A ramp rises 3 m over a horizontal distance of 4 m. tan θ = 3 ÷ 4 = 0.75. Use the inverse function on the calculator: θ = tan⁻¹(0.75) ≈ 36.9°.
Make sure your calculator is in degree mode. Some values are worth remembering: sin 30° = 0.5, cos 60° = 0.5 and tan 45° = 1.
Trigonometry only works directly in right-angled triangles. To find a missing side without an angle, use Pythagoras' theorem. Angles are measured as in angles, and solving for an unknown uses equations. Surveyors, pilots, engineers and game designers all use trigonometry.
Examples
| Known | Want | Use |
|---|---|---|
| angle and hypotenuse | opposite | sin |
| angle and hypotenuse | adjacent | cos |
| angle and adjacent | opposite | tan |
| two sides | the angle | inverse sin, cos or tan |
Common mistakes
- Mislabelling the sides. Opposite and adjacent change depending on which angle you choose.
- Using radian mode. For angles in degrees, the calculator must be in degree mode.
- Using trigonometry in a triangle without a right angle. That needs other rules.
Quick quiz
Trigonometry
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Related topics
Pythagoras' theorem
Pythagoras' theorem says that in a right-angled triangle, a² + b² = c², where c is the longest side (the hypotenuse). For sides 3 and 4, the hypotenuse is 5.
Angles
An angle measures the amount of turn between two lines that meet at a point. Angles are measured in degrees (°); a right angle is 90° and a full turn is 360°.
Equations
An equation says two things are equal, using an = sign, like 2x + 1 = 9. Solving it means finding the value of the letter that makes it true, here x = 4.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.
Area of a triangle
The area of a triangle is half of its base times its perpendicular height: A = ½ × b × h. A triangle with base 8 cm and height 5 cm has an area of 20 square centimetres.
Pi (π)
Pi (π) is the number you get when you divide any circle's circumference by its diameter. It is about 3.14, and its digits go on forever without repeating.