SINE PROBLEMS

ADVANCED

Concept: In a right triangle, the sine of the angle is what we call the ratio of the side opposite the angle and the hypotenuse.

Steps For Solving Sine Problems

Step 1 | Determine the length of the side which is opposite the angle in question.
Step 2 | Determine the length of the hypotenuse.
Step 3 | Set up a ratio using the length of the side opposite 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 sine of the angle in question.

Example 1 Solve for the unknown variable:

Step 1 | Determine the length of the side which is opposite the angle in question. | 70.711
Step 2 | Determine the length of the hypotenuse. | c
Step 3 | Set up a ratio using the length of the side opposite the angle as the numerator, and the hypotenuse as the denominator. (This ratio expresses the sine of the angle in question, which measures 45 ° in this problem.) | 70.711 c
| Since we were asked to solve for the variable, and since we have now accomplished that task, there is no need to continue with Steps 4 and 5. |

Example 2 A springboard is 40.15 yards above the sea. From the top of the springboard the angle of depression of a boat measures 35 degrees. How far is the boat from the top of the springboard?

NOTE: | We can solve this problem using a triangle. Use the line between the boat and the springboard as your hypotenuse. The line between the boat and the base of the springboard is the side adjacent the angle. The line between the base of the springboard and the springboard is the side opposite, and is 40.15 yards. The boat is the vertex of the angle in question, and that angle is 35 degrees. |
Step 1 | Determine the length of the side which is opposite the angle in question. | 40.15
Step 2 | Determine the length of the hypotenuse. | h
Step 3 | Set up a ratio using the length of the side opposite the angle as the numerator, and the hypotenuse as the denominator. (This ratio expresses the sine of the angle in question, which measures 35 ° in this problem.) | 40.15 . h
| We were asked to find the distance between the top of the springboard and the boat. In our triangle, that distance was represented by the hypotenuse. We have now determined that measurement to be 70 yards, so there is no need to continue with Steps 4 and 5. |