Finding an Angle

So far you have used an angle to find a side. Now work backwards. You know two sides and want the angle theta. From theta, five is opposite and ten is the hypotenuse, so sine theta is five over ten, which is one half. Now ask: which angle has a sine of one half? From the thirty, sixty, ninety triangle, sine thirty degrees is one half, so theta is thirty degrees. This backwards step is called the inverse sine, written sine to the power negative one, and it means the angle whose sine is.

Here the two short sides are four and four root three. From theta, four root three is opposite and four is adjacent, so use tangent: tan theta is four root three over four, which simplifies to root three. Which angle has a tangent of root three? Sixty degrees. You can also see it from the shape: short sides in the ratio one to root three belong to the thirty, sixty, ninety triangle, and the bigger angle is opposite the longer side. So theta is sixty degrees.

The hypotenuse is six root two and the side next to theta is six. Cosine theta is six over six root two, which is one over root two. That is the cosine of forty five degrees, so theta is forty five degrees. There is a second way to see it. Use Pythagoras for the third side: six root two squared is seventy two, minus thirty six leaves thirty six, so the third side is also six. When the two short sides are equal, the triangle is isosceles, and both acute angles are forty five degrees.

Most angles are not thirty, forty five or sixty. Then you need a table of values, and in a test you will be given one. From theta, six is opposite and ten is the hypotenuse, so sine theta is six divided by ten, which is zero point six. Now look down the sine column for zero point six. Sine thirty seven degrees is zero point six zero, so theta is about thirty seven degrees. Make sure you read the right column: if the table also lists cosines, the same number there belongs to a different angle.

Once you have one acute angle, the other one is free. The angles of a triangle add to one hundred and eighty degrees, and the right angle takes ninety, so the two acute angles add to ninety. Here sine theta is nine over eighteen, which is one half, so theta is thirty degrees. That leaves phi equals ninety minus thirty, which is sixty degrees. You can check it: from phi, the side of nine is adjacent, and cosine phi is nine over eighteen, one half, which is the cosine of sixty degrees. Quiz and exercises now.