Finding a Missing Angle with Inverse Tan

0:00This right triangle has sides of 5 centimetres and 8 centimetres. We want the angle marked theta.

0:08Name the sides from theta: the opposite side is across from it, the adjacent side is next to it, and the hypotenuse is the longest side.

0:19We know opposite and adjacent, so we use the tangent ratio: tan theta equals opposite over adjacent.

0:27Put in the lengths: 5 divided by 8 is 0.625.

0:33Tan turns an angle into a ratio. Inverse tan, written tan to the minus 1, works backwards: from a ratio to an angle.

0:44So theta equals inverse tan of 0.625.

0:49On most calculators, press shift or second, then tan, then 0.625, then equals.

0:58It shows 32.00538, so theta is about 32.0 degrees to one decimal place.

1:07Check: tan of 32.0 degrees is 0.6249, which is about 0.625.

1:17Now find the other acute angle, at the top. Acute means less than 90 degrees.

1:23The three angles add to 180 degrees. The right angle is 90, so the two acute angles add to 90.

1:32So the other angle is 90 minus 32.0, which is 58.0 degrees.

1:39Check: 32.0 plus 58.0 plus 90 is 180.

1:47Watch for two common mistakes. First, radians are another unit for angles. A calculator in radians mode gives 0.5586. Switch it to degrees to get 32.0.

2:01Second, pressing tan instead of inverse tan. In degree mode, tan of 0.625 is 0.0109, which is not the angle.

2:14To recap: find the ratio, then inverse tan turns it back into the angle.