COSINE PROBLEMS

ADVANCED

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

Steps For Solving Cosine Problems

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

Example 1 Solve for the unknown variable:

Step 1 | Determine the length of the side which is adjacent to the angle in question (and which is not the hypotenuse). | b
Step 2 | Determine the length of the hypotenuse. | 130
Step 3 | Set up a ratio using the length of the side adjacent to the angle as the numerator, and the hypotenuse as the denominator. (This ratio expresses the cosine of the angle in question, which measures 25 ° in this problem.) | b . 130
| We were asked to solve for the variable. We have now determined that measurement, so there is no need to continue with Steps 4 and 5. |

Example 2 How far from the base of a building is a ladder that measures 130 meters in length and is leaning against the building at a forty degree angle with the ground?

NOTE: | We can solve this problem using a triangle. Use the ladder as the hypotenuse. The base of the ladder is the vertex of the angle in question, which in this problem measures 45 degrees. The distance on the ground, from the base of the building to the base of the ladder is the side adjacent the angle in question, which is what we are trying to determine in this problem. |
Step 1 | Determine the length of the side which is adjacent to the angle in question (and which is not the hypotenuse). | b
Step 2 | Determine the length of the hypotenuse. | 130
Step 3 | Set up a ratio using the length of the side adjacent to the angle as the numerator, and the hypotenuse as the denominator. (This ratio expresses the cosine of the angle in question, which measures 25 ° in this problem.) | b . 130
| We were asked to find the distance between the base of the ladder and the base of the building. In our triangle, that distance was represented by the side adjacent to the angle in question. We have now determined that distance 92 meters, so there is no need to continue with Steps 4 and 5. |