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:
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.
Estimate the sin, cos and tan of 30°, using an accurate drawing of the triangle shown.
Here, hypotenuse = 11.6 cm , adjacent = 10 cm and opposite = 5.8 cm, so,
|cos30°||= =||0.86 (to 2 d.p.)|
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.
Measure the angle marked in the following triangle:
Calculate the sine, cosine and tangent of this angle.
|opposite||= 6 cm|
|adjacent||= 8 cm|
|hypotenuse||= 10 cm|