1. Module 7

1.26. Page 4

Mathematics 10-3 Module 7 Lesson 5

Module 7: Trigonometry

 

Bringing Ideas Together

 

In Explore you examined the ratio in right triangles of the side adjacent to the hypotenuse. This ratio is another example of a trigonometric function. You saw that, for similar right triangles and a particular acute angle, the ratio was the same value regardless of the size of the right triangle.

 

 

This illustration shows a right triangle ABC with angle A marked in red and angle C marked with a square. Side AC is labelled side adjacent and is aqua, side BC is labelled side opposite and is orange, and side AB is labelled hypotenuse and is black.

 

cosine ratio: the ratio of the length of the side adjacent to the length of the hypotenuse

This ratio is called the cosine ratio.

 

The cosine ratio is always in relation to an acute angle (in this case, ∠A). You must identify the acute angle when you write the cosine ratio in a particular situation.

 

So, cosine . It is often written more simply as .

 

Example 1

 

Draw a right triangle with an acute angle of 60°. Determine cosine 60° to two decimal places.

 

Solution

 

It does not matter how large you draw the triangle as all right triangles with a 60° angle will be similar in shape. However, a hypotenuse of 10 cm will make the calculation easier.

 

Once you draw the triangle, measure the adjacent side and hypotenuse to the nearest millimetre.

 

 

This illustration shows a right triangle with angle A between one leg and the hypotenuse measuring 60 degrees. The hypotenuse measures 10.0 centimetres, and the adjacent side measures 5.0 centimetres.

 

 

 

A ratio of 0.5 means that, in a right triangle with a 60° angle, the adjacent side will be one half the length of the hypotenuse. For instance, if the adjacent side were 30 in, the hypotenuse would be 60 in long, and .

 

Example 2

 

Determine cos 45°, correct to two decimal places, by measuring the sides of a right triangle.

 

Solution

 

Draw a suitable right triangle. You may wish to draw a right triangle with a hypotenuse of 10 cm to simplify the calculation. Then measure the adjacent side.

 

 

This illustration shows a right triangle with angle A between one leg and the hypotenuse measuring 45 degrees. The hypotenuse measures 10 centimetres and the adjacent side measures 7.1 centimetres.

 

 

 

Note: This answer is only approximate since measurements were only correct to the nearest millimetre.

 

This ratio means the adjacent side is shorter than the hypotenuse and is 0.71 times as long. So, if the hypotenuse were 20 cm long, the adjacent side would be about 20 × 0.71 cm = 14.2 cm long.

 

Self-Check

 

Use the method of the previous two examples to complete the following table. You may wish to use 10 cm for the hypotenuse.

 

SC 6. Complete the following table.

 

Angle

Hypotenuse

Side Opposite

Cosine

10°

 

 

cos 10° =

20°

 

 

cos 20° =

30°

 

 

cos 30° =

40°

 

 

 

45°

10 cm

7.1 cm

cos 45° = 0.71

50°

 

 

 

60°

10 cm

5.0 cm

cos 60° = 0.50

70°

 

 

 

80°

 

 

 

 

Compare your answers.

 

The Cosine Ratio

 

Look at the table of cosines you prepared. What do you notice about the value of the cosine ratio? When the angle is close to 0°, what happens to the cosine ratio as you move from 0° to 80°? How does this compare to what happens with the sine ratio?

 

As you have seen just seen and as their names suggest, cosine and sine ratios are related. You will now explore some other relationships that exist between the cosine and sine ratios.

 

Fill in the sine ratios in the table. What pattern can you see between the cosine and sine ratios?

 

Angle

Cosine

Angle

Sine

10°

0.98

80°

 

20°

0.94

70°

 

30°

0.87

60°

 

40°

0.77

50°

 

50°

0.64

40°

 

60°

0.50

30°

 

70°

0.34

20°

 

80°

0.17

10°

 

 

Consider the following right triangle.

 

 

This illustration shows right triangle PQR. The sides are labelled with lower case p, q, and r so that the lowercase p is opposite angle P, the lowercase q is opposite angle Q, and the lowercase r is opposite angle R. Side PR is drawn as a thick black line. Side PQ is a thin black line. Side QR is drawn as an orange line over a blue line. Angle R is coloured in blue, and angle P is coloured in orange.

 



 

So, cos R = sin P.

 

In Get Started you reviewed complementary angles. In the most recent triangle, ∠R and ∠P are complementary angles.

 

Therefore, the cosine of an angle is the sine of its complementary angle. In other words, the cosine of an angle and the sine of the angle’s complement match.

 

The cosine is just the sine of the complement angle. That’s where the name cosine comes from!


 

Just as for the sine ratio, you can use your calculator to determine the cosine ratio. Be sure your calculator is set to degree mode.

 

Example 3

 

Find the value of each of the following. Express each value so that the value is correct to 4 decimal places. Use your calculator.

  1. cos 45°
  2. Use your result from a. to find the sine ratio of the complement of 45°.
  3. cos 60°
  4. Use your result from c. to find the sine ratio of the complement of 60°.

Solution

  1. To find cos 45°, press this sequence of keys. If you do not obtain the same numbers in your calculator display, consult your manual. Calculators can vary in the order of key strokes.

    This illustration shows the key strokes to calculate the cosine of 45 degrees. The keys shown are in this order: Cos, 4, 5, right parenthesis, and the equals sign.

    You should see something like this.

    This illustration shows the calculator screen with the value of cosine 45 degrees. It is 0.7071067812. . .

    So, cos 45° 0.7071.
  1. The complement of 45° is also 45°, since 45° + 45° = 90°.

    So, sin 45° = cos 45° 0.7071.

  2. To find cos 60°, press this sequence of keys. If you do not obtain the answer shown, consult your manual.

    This illustration shows the keystrokes needed to fine cosine 60 degrees on a calculator. The keys are shown in this order, cos: 6, 0, right parenthesis, and the equal sign.

    You should see something like this.

    This illustration shows the calculator screen with the value of cosine 60 degrees. The value is 0.5.

    So, cos 60° = 0.5.

  3. The complement of 60° is 30°, since 60° + 30° = 90°.

    So, sin 30° = cos 60° = 0.5000.
Self-Check

 

Use your calculator to complete the following table.

 

SC 7. Complete this table. Round to four decimal places.

 

Angle

Cosine

10°

 

20°

 

30°

 

40°

 

45°

0.7071

50°

 

60°

0.5000

70°

 

80°

 

 

Compare your answers.

The sine ratio is always between 0 and 1. The cosine ratio decreases as the angle moves from 0° to 90°. But the sine ratio does the opposite–it increases as the angle moves from 0° to 90°.
For angles close to 0° in size, the side adjacent is almost as long as the hypotenuse, so the cosine ratio is close to 1 in value. As the angle increases in size, the side adjacent decreases in length in relation to the hypotenuse, so the cosine ratio decreases in value. For angles close to 90°, the cosine ratio is almost 0.
You can see from the table of cosines you prepared that, because the hypotenuse is always longer than the side adjacent, the cosine ratios for angles between 0° and 90° will be between 0 and 1.