Trigonometric ratios and angles of elevation

Last time

Before we start: last time, in Exact trigonometric values and trigonometric graphs, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

How tall is the tower?

You cannot climb a tower with a tape measure. But you can walk forty metres from its base, look up at the top, and measure the angle. With that one angle and one length, trigonometry gives the height. Three ratios, sine, cosine and tangent, do all the work, and you have already met right-angled triangles.

Naming the sides

In a right-angled triangle, the hypotenuse is the longest side, opposite the right angle. The opposite side is across from the angle you are using. The adjacent side is next to it, and is not the hypotenuse. Change the angle and opposite and adjacent swap. Always name the sides for the angle in the question.

SOH CAH TOA

Sine is opposite over hypotenuse. S O H. Cosine is adjacent over hypotenuse. C A H. Tangent is opposite over adjacent. T O A. Pick the ratio that uses the two sides in your problem.

Finding a side

A triangle has an angle of thirty-five and a hypotenuse of twelve. Find the side opposite the angle. Opposite and hypotenuse mean sine. Multiply both sides by twelve: x equals twelve times sine thirty-five. On the calculator, in degree mode, that is six point eight eight.