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Right Angled triangles

A right-angled triangle is any triangle with a right angle in it, such as this one:

A right angled triangle

a, b and c stand for the lengths of each side of the triangle. θ and φ are two of the angles. We know that the third angle (the one opposite side c) is a right angle, meaning that it is exactly 90° (or π/2, if you work in radians).

Suppose you know the value of either θ or φ, but not both. How do you work out the other angle?

In case you don't already know, the sum of all angles in a triangle is 180°. Always. What's more, this is true for all triangles, not just right-angled ones.

So:	θ + φ + 90°	=	180°
Therefore:	θ + φ	=	180° - 90°
		=	90°

Or, if you prefer to work in radians:	θ + φ + π/2	=	π
Therefore:	θ + φ	=	π/2

As you can see, if you know one of the two angles, it's easy to find the other one.

If you know the angles and two of the side lengths, how do you find the third side length?

This is where the sin, cos and tan functions come into play.

For any right-angled triangle, you can work out the side lengths from any of the following formulae:

sin(θ)	=	a / c
cos(θ)	=	b / c
tan(θ)	=	a / b

sin(φ)	=	b / c
cos(φ)	=	a / c
tan(φ)	=	b / a

Assuming you know θ (or φ), you can find the values of sin(θ), cos(θ) and tan(θ) using any scientific calculator. In most cases you can enter the angles into a scientific calculator in either degrees or radians - just set your calculator to DEG mode to work in degrees or RAD mode if you need to enter angles in radians instead.

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