Finding a missing side with a trig ratio

0:00A 12 metre ladder leans on a wall at 70 degrees to the ground. How high does it reach?

0:07The ladder is the hypotenuse: the longest side, across from the right angle.

0:13The height is the opposite side, across from the angle.

0:17The ground is the adjacent side: next to the angle, but not the hypotenuse.

0:23The three primary trig ratios: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent.

0:35Sine uses the opposite and the hypotenuse, so sine 70 degrees equals h over 12.

0:43h is on top, so multiply both sides by 12.

0:47Set your calculator to degrees. Sine 70 should show about 0.9397.

0:5512 times 0.9397 is 11.276, so h is about 11.3 metres up the wall.

1:05It is less than 12, as it must be: the hypotenuse is the longest side.

1:11New triangle: the angle is 40 degrees, the adjacent side is 10 metres. Find the hypotenuse, x.

1:19Cosine uses the adjacent and the hypotenuse: cosine 40 degrees equals 10 over x. This time, x is on the bottom.

1:29A common mistake is x equals 10 times cosine 40 degrees, which is 7.66: shorter than the adjacent side, so it cannot be right.

1:41Instead, multiply both sides by x, then divide by cosine 40 degrees: x equals 10 divided by cosine 40 degrees.

1:52On the calculator that is 13.054, so x is about 13.1 metres: longer than 10, as it should be.

2:02To recap: unknown on top, multiply. Unknown on the bottom, divide.