Definition: In a right triangle, the tangent of an acute angle is the length of the opposite side divided by the length of the adjacent side.
What does tangent measure?
For an acute angle A, tan A = opposite / adjacent. It compares the vertical rise with the horizontal run, so it is the slope of the hypotenuse’s line. A 45 degree angle has tangent 1.
When is tangent the best choice?
When the two sides involved are the two legs, and the hypotenuse is neither given nor required. Typical situations are the height of a tree from its shadow, or an angle of elevation.
Example. You stand 40 ft from a flagpole and look at its top with an angle of elevation of 54 degrees. Your eye is 5 ft off the ground. The height above eye level is 40 · tan 54 degrees. Since tan 54 degrees is about 1.3764, that is about 55.06 ft. The pole is about 60.1 ft tall.
What mistakes occur?
Forgetting to add the eye height, and mixing up this ratio with the tangent line of a circle, which is a different idea that shares the name. Tangent can be larger than 1. Finally, the inverse is tan-1, not the reciprocal 1 / tan.