Trigonometry to Find a Missing Side Length (Numerator). Step by Step Guide.

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In this article I will show you how to work out missing side lengths in a right angle triangle using trigonometry. All the examples here are when the missing side comes out on the numerator of the fraction.

To find a side length in a right angled triangle using trigonometry you will need to be given one of the angles in the triangle (other than the right angle!) and one of the side lengths.

Also, make sure you know how to label up you triangle:

The longest side is the hypotenuse (H)

The side opposite the given angle is called the opposite side (O)

The remaining side is called the adjacent side (A)

Example 1

Use trigonometry to work out the missing side length (?) in this right angled triangle.

Answer

First of all label up your triangle.

The sloping edge is H as this is the longest.

The height of the triangle is O as this side is opposite the 25⁰ angle.

That leaves the base of the triangle which will be called A.

Next, you need to select the correct trigonometric formula. There are 3 formulas to remember in trigonometry:

Sin Ѳ = 0/H

Cos Ѳ = A/H

Tan Ѳ = O/A

(Ѳ is the angle which you are given, in this case Ѳ = 25⁰)

Now in the question you have the Hypotenuse (this is 83mm) and you are trying to find the Adjacent side. Therefore, you need A and H in your formula, so the formula to use is Cos Ѳ = A/H

Now substitute the values into this trigonometric formula:

Cos Ѳ = A/H

Cos 25 = ?/83

Notice that the missing length (?) has come out on the top of the formula.

Finally, multiply both sides by 83.

83 × cos25 = ?

? = 75 mm to the nearest whole number.

Example 2

Use trigonometry to work out the missing side length (?) in this right angled triangle.

Answer

First of all label up your triangle.

The sloping edge is H as this is the longest side length.

The height of the triangle is O as this side is opposite the 58⁰ angle.

That leaves the base of the triangle which will be called A.

Again you need to select the correct trigonometric formula:

Sin Ѳ = 0/H

Cos Ѳ = A/H

Tan Ѳ = O/A

(Ѳ is the angle which you are given, in this case Ѳ = 25⁰)

Now in the question you have the Adjacent (this is 50mm) and you are trying to find the Opposite side. Therefore, you need A and O in your formula, so the formula to use is Tan Ѳ = O/A

Now substitute the values into this trigonometric formula:

Tan Ѳ = O/A

Tan 58 = ?/50

Notice again, that the missing length (?) has come out on the top of the formula.

Finally, multiply both sides by 50.

50 × tan58 = ?

? = 80 mm to the nearest whole number.

For some more worked question on working out sides in right angled triangles then click here.

For some real life question of working out side length’s  in right angled triangles then click here.

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