Introduction to trigonometry for right-angled triangles

Part ofMathsPythagoras and trigonometry

Key points

An image of a right angled triangle. The right angle is the bottom left vertex of the triangle. The bottom right vertex has been labelled theta. The base of the triangle has been labelled adjacent, a. The vertical side, opposite angle theta, has been labelled opposite, o. The diagonal side, opposite the right angle, has been labelled hypotenuse, h. The words, opposite, hypotenuse, and adjacent are coloured orange. The letters, o, h, and a are coloured blue. The triangle is coloured pink.
The sides of right-angled triangles are labelled opposite, hypotenuse and adjacent.

Trigonometry explores the relationship between sides and angles in right-angled triangles.

The sides of a right-angled triangle are labelled with:

  • Hypotenuse (\(h\)) – the longest side, which is opposite the right angle.
  • Opposite (\(o\)) – the side opposite to the given angle.
  • Adjacent (\(a\)) – the side next to the given angle.

The Greek letter Ɵ (theta) is often used as a symbol for an unknown (given) angle.

triangles are of each other. The angles in similar triangles are the same. The sides of similar triangles are in the same .

Trigonometric ratios show how long one side of the triangle is compared to another. The 3 important ratios are known as the sine (sin), cosine (cos) and tangent (tan) of the angle:

sin⁡Ɵ = \( \frac{opposite}{hypotenuse} \)

cos⁡Ɵ = \( \frac{adjacent}{hypotenuse} \)

tan⁡Ɵ = \( \frac{opposite}{adjacent} \)

An image of a right angled triangle. The right angle is the bottom left vertex of the triangle. The bottom right vertex has been labelled theta. The base of the triangle has been labelled adjacent, a. The vertical side, opposite angle theta, has been labelled opposite, o. The diagonal side, opposite the right angle, has been labelled hypotenuse, h. The words, opposite, hypotenuse, and adjacent are coloured orange. The letters, o, h, and a are coloured blue. The triangle is coloured pink.
The sides of right-angled triangles are labelled opposite, hypotenuse and adjacent.
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Labelling sides of a right-angled triangle for trigonometry

When a question about right-angled triangles involves all three sides and no angles, is used to calculate the value of a missing side.

When the question involves two sides and an angle in a right-angled triangle, is used.

The sides are labelled depending on where the angle is.

The sides are called the (\(h\)), the (\(o\)) side, and the (\(a\)) side.

The Greek letter theta, which has the symbol Ɵ, is often used to show an unknown angle.

Examples

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  1. Example one. An image of a right angled triangle. The right angle is the bottom left vertex of the triangle. The bottom right vertex has been labelled theta. A diagonal arrow points from the right angle to the diagonal side opposite. The side has been labelled, hypotenuse. The arrow and the word hypotenuse are coloured orange. The triangle is coloured pink.

    When using trigonometry in right-angled triangles, each side is labelled based on a given angle, Ɵ. The hypotenuse (𝒉) is the longest side of the triangle, which is always opposite the right angle.

Question

Label the right-angled triangle with the hypotenuse, opposite and adjacent sides.

An image of a right angled triangle. The right angle is the bottom right vertex of the triangle. The top right vertex has been labelled theta. The triangle is coloured pink.

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The three trigonometric ratios

Triangles that have the same three angles are of each other and are known as similar triangles.

For two right-angled triangles, if an angle (other than the right angle) is the same, the third angle will also be the same. This is true because angles in a triangle add up to 180°.

Therefore, right-angled triangles that include the same angle are .

For similar right-angled triangles, the , and sides stay in the same . For example, the adjacent side divided by the hypotenuse will give the same answer no matter how much the triangle is enlarged.

The hypotenuse, opposite and adjacent side can be divided in three important ways.

The answer to these division calculations depends on a given angle, Ɵ.

The three divisions are written as fractions. They are called the three trigonometric because they show how long one side is compared with another.

The trigonometric ratios are known as the sine (sin), cosine (cos) and tangent (tan) of the angle.

sin⁡Ɵ = \( \frac{opposite}{hypotenuse} \)

cos⁡Ɵ = \( \frac{adjacent}{hypotenuse} \)

tan⁡Ɵ = \( \frac{opposite}{adjacent} \)

Examples

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  1. An image of a square grid. The grid has a length of eighteen squares and a width of twelve squares. Two right angled triangles have been drawn on the grid. The first triangle has a base of length three squares, a height of length four squares and a hypotenuse of length five squares. The right angle is the bottom left vertex of the triangle. The length of each side is labelled. The angle marked theta is between the sides measuring four and five squares. The second triangle has a base of length six squares, a height of length eight squares and a hypotenuse of length ten squares. The right angle is the bottom left vertex of the triangle. The length of each side is labelled. The angle marked theta is between the sides measuring eight and ten squares. The first triangle is coloured green. The second triangle is coloured orange.

    The right-angled triangle on the left has sides 3, 4, and 5 units long. The right-angled triangle on the right has sides 6, 8, and 10 units long. The triangles are mathematically similar because one is an enlargement of the other. The lengths of the right triangle are double those on the left triangle

Question

Calculate the value of sinƟ, cosƟ and tanƟ for the right-angled triangle.
Give your answer as a fraction.

An image of a right angled triangle. The right angle is the top left vertex of the triangle. The triangle has a horizontal length of twenty eight metres, a vertical length of forty five metres and the diagonal length is fifty three metres. Angle theta is between the two sides measuring forty five metres and fifty three metres. The triangle is coloured green.

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Practise trigonometric ratios in right-angled triangles

Practise trigonometric ratios in similar right angled triangles with this quiz. You may need a pen and paper to help you with your answers.

Quiz

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