Trigonometry in the Right Triangle · Grade 10

Why does the ratio between sides depend only on the angle?

Two right triangles sharing an acute angle are similar, and in similar triangles the ratio between any two corresponding sides is equal. So the ratio does not depend on the size of the triangle but only on the angle — which is why it can be written in a single table and used for every triangle.

Learning objectives

Same angle, same ratio

Take a right triangle with an acute angle of thirty degrees and double its size. Every side doubles, so the ratio between any two sides stays exactly the same number.

That is no accident. Two right triangles sharing an acute angle are similar, because the third angle is then fixed — the angles of a triangle add to a hundred and eighty. And in similar triangles every ratio between corresponding sides is equal.

What this makes possible

Since the ratio depends only on the angle, it can be worked out once for each angle and written in a table. That is exactly what a calculator does when you press sine: it returns a ratio, not a length.

It also means there is no point asking which triangle. Thirty degrees in a tiny triangle and thirty degrees in a triangle the size of a building give the same number.

Naming the sides

The hypotenuse is always the side facing the right angle, and it is the longest. The other two are named relative to the angle you are talking about: the side opposite it and the side adjacent to it.

Note that opposite and adjacent swap when you move to the other acute angle in the same triangle. That is why it pays to mark the angle on the diagram before writing any ratio at all.

Worked examples

  1. In a right triangle the side opposite a 30 degree angle is 3 and the hypotenuse is 6. What is the ratio?

    1. Identify the sides: the opposite side is 3, the hypotenuse is 6
    2. The ratio wanted is opposite over hypotenuse
    3. Divide: 3 over 6

    Answer: The ratio is 0.5

  2. A similar triangle is four times bigger: the opposite side is 12 and the hypotenuse 24. What is the ratio?

    1. Both sides grew by a factor of four, so the triangle is similar to the first
    2. Write exactly the same ratio: opposite over hypotenuse
    3. Divide: 12 over 24

    Answer: 0.5 again — the size changed nothing

  3. One acute angle in a right triangle is 40 degrees. What is the other?

    1. The angles of a triangle add to 180
    2. One of them is right, that is 90
    3. That leaves 90 for the two acute angles together

    Answer: The other angle is 50 degrees

Common mistakes

Looking for the ratio by side lengths rather than by the angle
The ratio is fixed by the angle alone. Two triangles with the same acute angle, at any size, give exactly the same number.
Confusing the opposite side with the adjacent one
Opposite and adjacent are defined relative to the angle you are discussing, and they swap when you move to the other acute angle. Marking the angle on the diagram before writing the ratio prevents it.
Treating the hypotenuse as an ordinary side
The hypotenuse always faces the right angle and is always the longest. If a sine or cosine comes out greater than one, you have divided by the wrong side.

What to remember

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