4.2 The Trigonometrical Ratios Introduction The trigonometrical ratios sine, cosine and tangent appear frequently in many engineering problems. This leaflet revises the meaning of these terms. 1. Sine, cosine and tangent ratios Study the right-angled triangle ABC shown below. C hypotenuse side opposite to θ θ A B side adjacent tο θ The side opposite the right-angle is called the hypotenuse. The side opposite to θ is BC. The remaining side, AB, is said to be adjacent to θ. Suppose we know the lengths of each of the sides as in the figure below. C 10 8 θ A 6 B We can then divide the length of one side by the length of one of the other sides. The ratio BC is known as the sine of angle θ. This is abbreviated to sin θ. In the triangle shown AC we see that 8 = 0.8 sin θ = 10 is known as the cosine of angle θ. This is abbreviated to cos θ. In the triangle The ratio AB AC shown we see that 6 cos θ = = 0.6 10 is known as the tangent of angle θ. This is abbreviated to tan θ. In the triangle The ratio BC AB shown we see that 8 tan θ = = 1.3333 6 In any right-angled triangle we define the trigonometrical ratios as follows: www.mathcentre.ac.uk 4.2.1 c Pearson Education Ltd 2000 sin θ = BC opposite = hypotenuse AC cos θ = tan θ = opposite BC = adjacent AB adjacent AB = hypotenuse AC 2. Some standard, or common, triangles 2 √2 1 45° 30° √3 1 1 sin 45◦ = √ , 2 1 sin 30◦ = , 2 √ 3 ◦ , sin 60 = 2 60° 1 1 cos 45◦ = √ , 2 √ 3 cos 30◦ = , 2 1 cos 60◦ = , 2 tan 45◦ = 1 1 tan 30◦ = √ 3 √ tan 60◦ = 3 3. Using a calculator If we know the angles in a right-angled triangle the trigonometrical ratios can be found using a scientific calculator. Look for the sine, cosine and tangent buttons on your calculator and make sure that you can use them by verifying that sin 50◦ = 0.7660, cos 32◦ = 0.8480 Your calculator will be able to handle angles measured in either radians or degrees. It will be necessary for you to choose the appropriate units. Study your calculator manual to learn how to do this. Check that sin 0.56 radians = 0.5312, tan 1.4 radians = 5.7979 4. Finding an angle when a trigonometrical ratio is known If we are given, or know, a value for sin θ, cos θ or tan θ we may want to work out the corresponding angle θ. This process is known as finding the inverse sine, inverse cosine or inverse tangent. Your calculator will be pre-programmed for doing this. The buttons will be labelled invsin, or sin−1 , and so on. Check that you can use your calculator to show that if sin θ = 0.75 then θ = 48.59◦. Mathematically we write this as follows: if sin θ = 0.75, then θ = sin−1 0.75 = 48.59◦ www.mathcentre.ac.uk 4.2.2 c Pearson Education Ltd 2000

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