TANGENT PROBLEMS

ADVANCED

Concept: In a right triangle, the tangent of the angle is what we call the ratio of the side opposite the angle and the side adjacent to the angle.

Steps For Solving Tangent Problems

Step 1 | Determine the length of the side which is opposite the angle in question.
Step 2 | Determine the length of the side which is adjacent to the angle in question (and which is not the hypotenuse).
Step 3 | Set up a ratio using the length of the side opposite the angle as the numerator, and the length of the side adjacent to the angle as the denominator.
Step 4 | Divide the numerator by the denominator.
Step 5 | The quotient is the tangent of the angle in question.

Example 1 Solve for the tangent of the following angle:

Step 1 | Determine the length of the side which is opposite the angle in question. | 107.246
Step 2 | Determine the length of the side which is adjacent to the angle in question (and which is not the hypotenuse). |
| To determine the length of the adjacent side we will use the Pythagorean theorem. | a2 + b2 = c2 107.2462 + b2 = 1402 11500.42 + b2 = 19600 -11500.42 -11500.42 b2 = 8099.58 b = 90
Step 3 | Set up a ratio using the length of the side opposite the angle as the numerator, and the length of the side adjacent to the angle as the denominator. | 107.246 90
Step 4 | Divide the numerator by the denominator. |
Step 5 | The quotient is the tangent of the angle in question. | 1.191

Example 2 Ironman is flying at an altitude of 15.628 meters. He spots a lost button on the ground at an angle of depression of 10 degrees. How far is the lost button, horizontally, from Ironman?

NOTE: | We can solve this problem using a triangle. We will let the line from Ironman to the ground represent the side adjacent to the angle. The line from Ironman to the button will represent the hypotenuse of the right triangle. The angle formed by these two lines is the angle in question and in this problem it measures 10 degrees. We must determine the measure of the side opposite the angle in question, which is the HORIZONTAL distance between Ironman and the button. |
Step 1 | Determine the length of the side which is opposite the angle in question. | a
Step 2 | Determine the length of the side which is adjacent to the angle in question (and which is not the hypotenuse). | 15.628
Step 3 | Set up a ratio using the length of the side opposite the angle as the numerator, and the length of the side adjacent to the angle as the denominator. | a .. 15.628
| We were asked to find the HORIZONTAL distance between the button and Ironman. In our triangle, that distance was represented by the side opposite the angle in question. We have now determined that measurement to be 2.755 meters, so there is no need to continue with Steps 4 and 5. |