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Finding unknown sides and angles

Finding unknown sides and angles in a right-angled triangle can be done relatively easily. In order to do that, we first need to find the trigonometric ratio involved from any of the given sides or angles, and then calculate the unknowns.

Here are 3 examples of finding unknown sides and angles:

 

Example 1: To find the dimension of the opposite side.

calculating the side of the triangle

In the above figure, we need to find x (the opposite side), when the adjacent side = 80 m, and the angle involved is 38°.

The trigonometric ratio involved here is tan, and it is represented as –

tan 38° =  {opposite side}/{adjacent side}

= x/80

x = 80 x tan 38°

=  62.5028

=   62.50 m (to nearest 2 decimal places)

 

Example 2: To find the dimension of the hypotenuse.

calculating the hypotenuse of the triangle

In the figure above, we need to find x (the hypotenuse), when the adjacent side = 20 m, and the angle involved is 60°.

The trigonometric ratio involved here is cos, and it is represented as –

cos 60°  =  {adjacent side}/{hypotenuse}

=  20/x

x  = 20/{cos 60°}

= 40 m

Hint: When the unknown quantity is in the denominator, swap the positions across; here swap the positions of x and cos 60°.

 

Example 3: To find the angle of the triangle when the dimensions of the sides are given.

calculating the unknown angle of the triangle

In the above figure, we need to find the angle, when the opposite side = 3 m, and the hypotenuse = 5 m.

The trigonometric ratio involved here is sin, and it is represented as –

sin θ = {opposite side}/{hypotenuse}~=~3/5

θ = sin^-1(3/5)

= 36.8698

= 36° 52′ (to the nearest minute)