Mathematics for science and technology
Mathematics for science and technology

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Mathematics for science and technology

2 Sine, cosine and tangent

Sin, cos and tan are the commonly used abbreviations for each function. So:

  • Sin = sine
  • Cos = cosine
  • Tan = tangent

Sine is pronounced as ‘sign’ and cosine as ‘co-sign’.

Based on the triangle in Figure 2 these functions are defined as follows:

A right angle triangle labelled with ‘adjacent’, ‘opposite’ and ‘hypotenuse’.
Figure 2 A right angle triangle
sine of beta equals opposite open b close divided by hypotenuse open c close
cosine of beta equals adjacent open a close divided by hypotenuse open c close
tangent of beta equals opposite open b close divided by adjacent open a close

There is an easy way to remember this: SOHCAHTOA (pronounced sock -ah- toe-a).

        cap s in of equals cap o divided by cap h  Cos equals cap a divided by cap h  Tan equals cap o divided by cap a

Using these functions, and Pythagoras theorem, if you know two sides or a side and an angle for a right-angled triangle, then you can find the other side, sides or angle as required.

Activity 2 Sine and cosine values between 0° and 90°

Timing: Allow about 5 minutes

Making sure that your calculator is still set to degrees, work out these values for sine and cosine.

1.Sin 0°
2.Cos 0°
3.Sin 10°
4.Cos 10°
5.Sin 25°
6.Cos 25°
7.Sin 45°
8.Cos 45°
9.Sin 70°
10.Cos 70°
11.Sin 90°
12.Cos 90°

Do you notice anything interesting about your answers?

Answer

Angle/degreesSineCosine
001
100.17360.9848
250.42260.9063
450.70710.7071
700.93970.3420
9010

Perhaps you noticed that all the values for sine and cosine seem to lie between 0 and 1 for angles between 0° and 90°.

You may have also observed that as the values for sine increase, the values for cosine decrease.

In fact, for any angle between 0° and 90° the values for sine and cosine lie between 0 and 1.

These two graphs (Figures 3 and 4) show the shape of the sine and cosine functions between 0 and 360°. From this you can see that both functions stay within –1 and +1.

The sine function, shown in graph form.
Figure 3 The sine function
The cosine function, shown in graph form.
Figure 4 The cosine function

As shown in Figure 5 the tangent function is not similar in the way that cosine and sine are but forms its own repeating pattern.

The tangent function, shown in graph form.
Figure 5 The tangent function

Now it is time to see how you can use these functions with right-angled triangles.

MST_1

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