Finding a Side

To find a side, first label the triangle from the angle you are given, here thirty five degrees. Now ask two questions: which side do I know, and which side do I want? You know the hypotenuse, twenty. You want x, which touches the angle and is not the hypotenuse, so it is the adjacent side. Adjacent and hypotenuse appear together in C A H, so you use cosine: cos thirty five degrees equals x over twenty. The opposite side is not involved, so ignore it.

Here the angle is forty degrees, the hypotenuse is ten, and x is opposite the angle. Opposite and hypotenuse means sine: sine forty degrees equals x over ten. The unknown x is on the top of the fraction, so multiply both sides by ten: x equals ten times sine forty degrees. You are told sine forty degrees is about zero point six four, so x is about six point four. Always check that your answer is shorter than the hypotenuse.

Sometimes the unknown ends up on the bottom. The angle is fifty degrees, the opposite side is nine and x is adjacent, so use tangent: tan fifty degrees equals nine over x. Multiply both sides by x to get x times tan fifty degrees equals nine. Then divide by tan fifty degrees: x equals nine divided by tan fifty degrees. With tan fifty degrees about one point two, x is nine divided by one point two, which is seven point five.

Two triangles give exact values, so you never need a calculator for thirty, forty five or sixty degrees. Cut an equilateral triangle of side two in half and you get a thirty sixty ninety triangle with sides one, root three and two. The shortest side, one, is opposite thirty degrees. Cut a square of side one along its diagonal and you get a forty five degree triangle with sides one, one and root two. Read the ratios straight off: for example sine thirty degrees is one half, and tan sixty degrees is root three.

Put the exact values to work. The hypotenuse is eight and the angle is sixty degrees. The adjacent side x uses cosine: x equals eight times cos sixty degrees, which is eight times one half, so four. The opposite side y uses sine: y equals eight times sine sixty degrees, which is eight times root three over two, so four root three. You can also see it in one go: the sides are in the ratio one to root three to two, and here two has become eight, so everything is scaled by four. Quiz and exercises now.