medium difficulty6 min

Trigonometric ratios

Trigonometric ratios are the fixed ratios between the sides of a right triangle for a chosen acute angle.

In one line: Think: opposite is across, adjacent is beside, hypotenuse is the longest side.

Understand it

For one acute angle in a right triangle, the sides keep a fixed proportion. That is why ratios like sine, cosine, and tangent depend on the angle, not on the size of the triangle. If two right triangles have the same acute angle, the ratios for that angle stay the same.

Example

If two right triangles both have angle A equal to 30 degrees, then the ratio opposite side divided by hypotenuse for angle A will be the same in both triangles.

Formula and rule

For a right triangle with acute angle A: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent.

These are ratios of sides.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1In a right triangle, if the side opposite angle A is 3 cm and the hypotenuse is 5 cm, then sin A = 3/5.
2The ratio is a number between 0 and 1 for an acute angle.

Method you can reuse

1Mark the given angle.
2Find the right angle and identify the hypotenuse.
3Name the opposite and adjacent sides with respect to the given angle.
4Substitute the correct sides into the required ratio.

Sketch it in your answer

Draw a right triangle, mark one acute angle as A, label the side opposite A, the side adjacent to A, and the hypotenuse opposite the right angle.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why are trigonometric ratios called ratios and not side lengths?

Think first, then show answer

They are called ratios because each one compares two side lengths, such as opposite divided by hypotenuse or opposite divided by adjacent. The value depends on the angle, not on the actual size of the triangle.

Common wrong answer

A common wrong answer is to use the same side names for every angle.

Why it loses marks: Opposite and adjacent change when the chosen angle changes.

Don't mix this up

Students often mix up opposite side and adjacent side, especially when the triangle is drawn in a different orientation.

Exam tip

Always identify the chosen angle first, then label opposite, adjacent, and hypotenuse before writing any ratio.