1. Module 7

1.16. Page 6

Mathematics 10-3 Module 7 Lesson 3

Module 7: Trigonometry

 

Solving Problems Using the Sine Ratio

 

In the next example you will apply your calculator skills to help you solve a problem involving the sine ratio.

 

This photograph is of a mountain slope prone to avalanches.

iStockphoto/Thinkstock

Example 5

 

The photograph is of a mountain slope that often has avalanches.

 

Avalanches are triggered by human activity, such as snowmobiling, and most commonly occur on slopes between 35° and 45°.

 

Suppose the slope of a mountain slope is 38°, and the slope passes through a height of 800 m. What is the slope’s length? Round your answer to the nearest 100 m.

 

Solution

 

Draw a diagram.

 

 

This illustration shows a right triangle with angle A labelled and the Opposite side labelled with length 800 metres. The hypotenuse is labelled with length x.

 

Let x be the length of the slope.

 

 

 

The mountain slope would be approximately 1300 m in length.

 

Self-Check

 

Use the method outlined in Example 5 to solve this question.

 

SC 7. Solve for x. Round to two decimal places.

 

 

This illustration shows right triangle ABC with angle C marked as a right angle. Angle A measures 25 degrees, and the hypotenuse is 40 centimetres long. The opposite side is of measure x.

 

Compare your answers.

 

Finding Angles Using the Sine Ratio

 

You have just seen how an unknown side in a right triangle can be determined using the sine ratio, provided you know the measure of an acute angle in that triangle. Can you find an angle using the sine ratio?

 

For instance, suppose you are told that the sine of a given acute angle in a right triangle is 0.6. What is the angle?

 

Example 6

 

If sin A = 0.6, determine ∠A to the nearest degree.

 

View the animated “Sine Solution.”

 

Self-Check

 

Practise finding angles given the sine ratio.

 

SC 8. Find ∠A given sin A = 0.8.

 

Use both methods as shown in Example 6 to find the angle.

 

SC 9. Use your calculator to find each angle from its sine. Round your answers to the nearest tenth.

 

sin A

∠A

0.1257

 

0.7826

 

0.9000

 

 

 

 

SC 10. Find ∠A to the nearest tenth of a degree.

 

 

This illustration shows a right triangle with a non-right angle labelled A. The hypotenuse is 7 centimetres long, and the opposite side is 2 centimetres long.

 

Compare your answers.

 

Mastering Concepts

 

Try these questions. When you are finished, check your answer.

 

 

This illustration shows right triangle PQR with the sides labelled. Side PQ, which is opposite angle R, is labelled r. Side PR, which is opposite angle Q, is labelled q. Side QR, which is opposite angle P, is labelled p.

  1. For ΔPQR, write the ratios for sin P and sin R.
  2. How are ∠P and ∠R related?
  3. How are p and r related?

Compare your answers.