1. Module 7

1.15. Page 5

Mathematics 10-3 Module 7 Lesson 3

Module 7: Trigonometry

 

The Sine Ratio

 

Look at the table of sine ratios you prepared. What do you notice about the value of the sine ratio? What happens to the sine ratio as you move from an angle of 0° to 80°?

 

You should remember two benchmarks: sin 30° = 0.50, and sin 45° = 0.71.

 

These benchmarks, or referents, will help you assess whether or not an answer is reasonable!

 

Did You Know?

 

People tend to think of mathematics as a modern way of looking at the world. Did you know that the ancient Babylonians, Egyptians, Indians, and Greeks used trigonometric tables, especially in astronomy? The word sine, in fact, is a mistranslation into Latin. The word sinus means “bay,” which is what the translators thought the Arabic word written as jb meant. In fact, jb was simply borrowed from the Sanscrit (Indian) word jya-ardha, which means “half chord.” Indian mathematicians from more than 1500 years ago calculated tables of sines from half chords of circles correct to 4 decimal places!



This illustration shows a circle with centre C and chord AB. Segments are drawn from A to C and from B to C to form triangle ABC. A perpendicular bisector of AB is drawn from C. The bisector meets AB at point X.

A chord is a line segment joining two points on a circle. Segment AB is a chord, and segment AX is a half chord. The . Fortunately, you don’t have to calculate sines using half chords as early mathematicians did. You can use a calculator!

 

Example 4

 

Find each sine, correct to 4 decimal places, using your calculator.

  1. sin 45°
  2. sin 30°

Solution

 

There are several different ways angles may be expressed. Make sure your calculator is set to degree mode.

  1. To find sin 45°, press this sequence of keys. If you don’t obtain the answer shown, consult your manual.

     
    This illustration shows the key strokes required to calculate the sine of 45 degrees on a calculator. The keys shown are Sin, 4, 5, right parenthesis, and equals.

    Your calculator should display something similar to what follows.

     
    This illustration shows the calculator screen with the value of sine 45 degrees. The value shown is 0.7071067812 . . .

    So, sin 30° 0.7071.

  2. To find sin 30°, press this sequence of keys. If you don’t obtain the answer shown, consult your manual.

     
    This illustration shows the key strokes to calculate sine 30 degrees on a calculator. The keys shown are Sin, 3, 0, right parenthesis, and equals.

    Your calculator should display something similar to what follows.

     
    This illustration shows the calculator screen with the value of sine 45 degrees. The value shown is 0.5.

    So, sin 30° = 0.5.
Self-Check

 

Use the calculator method of the previous examples to complete the following table.

 

SC 6. Complete the following table. Round to 4 decimal places.

 

Angle

Sine

10°

 

20°

 

30°

0.5000

40°

 

45°

 

50°

 

60°

 

70°

 

80°

 

 

Compare your answers.

The sine ratio is always between 0 and 1. For angles close to 0° in size, the sine ratio is close to 0 in value. As the angle increases in size, the side opposite increases in length, so the sine ratio increases in value. For angles close to 90°, the side opposite is nearly as long as the hypotenuse, so the sine ratio is almost 1.