Unit 15 Section 2 : Trigonometric Functions

In this section we introduce 3 functions: sine, cosine and tangent, and their use in right-angled triangles. First we look at the conventions used for the names of the sides of a right-angled triangle with respect to one of the angles.

The adjacent side is the side joining the angle and the right angle.

The opposite side is opposite the angle.

The hypotenuse is the side opposite the right angle and is the longest side in the triangle.

Using these definitions, we can write down the trigonometric functions:

sinΘ   =     =  
cosΘ   =     =  
tanΘ   =     =  

Note that we abbreviate sine, cosine and tangent to sin, cos and tan.

In the following Examples and Exercises, we investigate the properties of these trigonometric functions.

Example 1

Estimate the sin, cos and tan of 30°, using an accurate drawing of the triangle shown.

The triangle has been drawn accurately below, and the sides measured.

Here, hypotenuse = 11.6 cm , adjacent = 10 cm and opposite = 5.8 cm, so,

sin30° = = 0.5
cos30° = = 0.86 (to 2 d.p.)
tan30° = = 0.58

Note that if we had drawn a similar right-angled triangle, again containing the 30° angle but with different side lengths, then we may have obtained slightly different values for sin30°, cos30° and tan30°. You can obtain more accurate values of sin30°, cos30° and tan30° by using a scientific calculator. If you have a calculator with the trigonometric functions, do this and compare them with the values above.

WARNING: When you use a scientific calculator, always check that it is dealing with angles in degree mode.

Example 2

(a)

Measure the angle marked in the following triangle:

In this case the angle can be measured with a protractor as 37°.
(b)

Calculate the sine, cosine and tangent of this angle.

Here we have
opposite = 6 cm
adjacent = 8 cm
hypotenuse = 10 cm
sinΘ = = = 0.6
cosΘ = = = 0.8
tanΘ = = = 0.75

Exercises

Question 1

Draw 3 different right-angled triangles that each contain a 60° angle.

(a)

Use each triangle to estimate sin60°, and check that you get approximately the same value in each case.

(b)

Estimate a value for cos60° .

(c)

Estimate a value for tan60° .

Question 2

Draw a right-angled triangle that contains an angle of 50°.

Use this triangle to estimate:

cos50°,
sin50°,
tan50°,
Question 3

Draw a right-angled triangle which contains a 45° angle.

State the value of tan45°.

tan 45° =

sin 45° = cos 45° because the triangle is isosceles so the opposite and adjacent have the same length. This means that tan45° = 1.
Question 4

Copy and complete the following table, giving your values correct to 2 significant figures. Draw appropriate right-angled triangles to be able to estimate the values.

Anglesinecosinetangent
10°
20°
30°
40°
50°
60°
70°
80°
Use the sin, cos and tan keys on your calculator to check your values.
Question 5

A pupil states that the sine of an angle is 0.5. What is the angle?

The angle is °.
Question 6

If the cosine of an angle is 0.17, what is the angle? Give the most accurate answer you can obtain from your calculator and then round it to the nearest degree.

The angle is °.
Question 7

What are the values of:

(a)cos 0°
(b)sin 0°
(c)sin 90°
(d)cos 90°
(e)tan 0°
(f)tan 90°
Question 8

Use your calculator to obtain the following, correct to 3 significant figures:

(a)sin 82°
(b)cos 11°
(c)sin 42°
(d)tan 80°
(e)tan 52°
(f)tan 38°
Question 9

Use your calculator to obtain the angle Θ , correct to 1 decimal place, if:

(a)cos Θ = 0.3Θ = °
(b)sin Θ = 0.77Θ = °
(c)tan Θ = 1.62Θ = °
(d)sin Θ = 0.31Θ = °
(e)cos Θ = 0.89Θ = °
(f)tan Θ = 11.4Θ = °
Question 10

A student calculates that cosΘ = 0.8.

(a)

By considering the sides of a suitable right-angled triangle, determine the values of sinΘ and tanΘ

sin Θ =
tan Θ =
The most obvious triangle is a 6, 8, 10 right-angled triangle, or its enlargements.
(b)

Use a calculator to find the angle Θ .

Θ = ° (to 1 d.p.)
(c)

Use the angle you found in part (b) to verify your answers to part (a).

sin Θ =
tan Θ =