Introduction to trigonometry for right-angled triangles

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. — Image caption,
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 = \( \frac{opposite}{adjacent} \)

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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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.
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The opposite side (𝒐) is the side opposite the given angle.
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The adjacent side (𝒂) is the side next to the given angle. It is between the given angle and the right angle.
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A fully labelled triangle.
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Label the right-angled triangle with the hypotenuse, opposite and adjacent side.
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The hypotenuse is the longest side and is always opposite the right angle. The opposite side is the side opposite the given angle. The adjacent side is the side next to the given angle. It is between the given angle and 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.

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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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
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The sides have been labelled as the hypotenuse (𝒉), opposite (𝒐), and adjacent (𝒂) based on the angle Ɵ.
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The opposite side divided by the hypotenuse for each triangle is 3/5 and 6/10. These two fractions are equivalent and both equal 0·6. No matter how much the triangle is enlarged, the opposite divided by the hypotenuse will always be 0·6. This is because the angle stays the same and the sides stay in the same proportion.
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The answer to the opposite side divided by the hypotenuse depends on how big the angle is. The opposite side divided by the hypotenuse is called the sine of the angle, and is abbreviated to sin. For a given angle Ɵ, sinƟ = opposite ÷ hypotenuse. This is written as a fraction.
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The adjacent side divided by the hypotenuse for each triangle is 4/5 or 8/10. These two fractions are equivalent and both equal 0·8. No matter how much the triangle is enlarged, the adjacent divided by the hypotenuse will always be 0·8. This is because the angle stays the same and the sides stay in the same proportion.
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The answer to the adjacent side divided by the hypotenuse depends on how big the angle is. The adjacent side divided by the hypotenuse is called the cosine of the angle, and is abbreviated to cos. For a given angle Ɵ, cosƟ = adjacent ÷ hypotenuse. This is written as a fraction.
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The opposite divided by the adjacent for each triangle is 3/4 or 6/8. These two fractions are equivalent and both equal 0·75. No matter how much the triangle is enlarged, the opposite divided by the adjacent will always be 0·75. This is because the angle stays the same and the sides stay in the same proportion.
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The answer to the opposite side divided by the adjacent depends on how big the angle is. The opposite side divided by the adjacent side is called the tangent of the angle and is abbreviated to tan. For a given angle Ɵ, tanƟ = opposite ÷ adjacent. This is written as a fraction.
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These three equations are true for all right-angled triangles. They are known as the three trigonometric ratios because they show how large one side is compared to the other based on an angle Ɵ.