As we know, the three basic trigonometric functions are as follows: Figure 1
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1 Trigonometry Basic Functions As we know, the three basic trigonometric functions are as follows: sin θ = cos θ = opposite hypotenuse adjacent hypotenuse tan θ = opposite adjacent Where θ represents an angle in a right-angled triangle. Worked examples: Figure 1 In Figure 1 above, find the value of the following to two decimal places: 1. sin α 2. cos α 3. tan α
2 Working: 1. sin α As we know, the sine of an angle is the opposite side over the hypotenuse, therefore: sin α = opposite hypotenuse sin α = sin α = sin α = 0.67 Here we plug the values from the triangle into our formula We now compute the answer And finally, we round it off as instructed 2. cos α The cosine of an angle is the adjacent side over the hypotenuse. Thus: cos α = adjacent hypotenuse cos α = cos α = cos α = 0.74 Again, we plug the values in, Compute the answer And round it off 3. tan α The tangent ratio of an angle is the opposite side over the adjacent side. Therefore: tan α = opposite adjacent tan α = tan α = tan α = 0.90 We plug in the values from the triangle, Compute the answer, And round it off. Now, it s perfectly possible that you will be asked to use these ratios to find the value of a side of the triangle. This is not particularly difficult; all we need to do is set up an equation and solve it just like any other.
3 Worked Example: Figure 2 In Figure 2 above, we need to find the value of the side x correct to 4 decimal places. In order to solve this, we need to look at what we have and what we want; we have the angle 20 and we know the value of one side, 65. Ask yourself, which trigonometric ratio includes everything we know and what we want to find? sine includes the angle and side we know (opposite) and the side we don t know (hypotenuse) sin θ = opposite hypotenuse We plug our values in; we now need to get x on its own sin 20 = 65 x We multiply both sides by x x. sin 20 = 65 And now divide both sides by sin 20 x = 65 sin 20 This can be worked out on your calculator x = And rounded as instructed x =
4 Exercise: Figure 3 1. In Figure 3 above, find the value of y correct to 3 decimal places. Show all your working. (5) Figure 4 2. In Figure 4 above, find the values of both r and s correct to the nearest whole number. Show all your working. (10)
5 Reciprocal Functions In addition to the three basic trigonometric functions, there are what are known as reciprocal functions. These are called cosecant, secant and cotangent and are defined as follows: cosec θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ For any right-angled triangle, these reciprocals may also be defined in terms of the sides of the triangle: cosec θ = hypotenuse opposite sec θ = hypotenuse adjacent cot θ = adjacent opposite You may have noticed that these are the opposite of the corresponding basic trig function. This is what makes them reciprocals and is why: sin θ cosec θ = 1 cos θ sec θ = 1 tan θ cot θ = 1 Your calculator is unlikely to have a button for these reciprocal functions. Therefore, in order to calculate them you will need to calculate the corresponding basic function and divide 1 by it. Worked Example: Figure 5
6 In Figure 5 above, calculate the following correct to 3 decimal places: 1. cosec α 2. sec α 3. cot β Working: 1. cosec α The cosecant of an angle may be defined thus: The cosecant is the reciprocal of the sine: cosec θ = 1 sin θ We are looking at angle α: cosec α = 1 sin α The sine of angle α is opposite over hypotenuse: cosec α = Which simplifies as below (note that this is the same as hypotenuse over opposite): cosec α = Which we can calculate as equal to: cosec α = And round as instructed: cosec α = sec α The secant of an angle can be calculated in the following way: The secant is the reciprocal of the cosine: sec θ = 1 cos θ We are looking at angle α: sec α = 1 cos α
7 The cosine of angle α is adjacent over hypotenuse: sec α = Which simplifies as below (note that this is the same as hypotenuse over adjacent): sec α = Which we can calculate as equal to: sec α = And round as instructed: sec α = cot β The cotangent of an angle can be calculated in the following way: The cotangent is the reciprocal of the tangent: cot θ = 1 tan θ We are looking at angle β: cot β = 1 tan β The tangent of angle β is opposite over adjacent: cot β = Which simplifies as below (note that this is the same as adjacent over opposite): cot β = Which we can calculate as equal to: cot β = And round as instructed: cot β = 0.903
8 Exercise: 1. Use your calculator to calculate the following correct to 2 decimal places: 1.1. cos sin tan sin cos For each of the following triangles, state whether a, b, or c are the hypotenuse, opposite or adjacent sides with respect to the angle θ: If x = 45 and y = 39 use your calculator to determine whether or not the following statements are true or false: 3.1. cos x + 2 cos x = 3 cos x 3.2. cos 2y = cos y + cos y sin x 3.3. tan x = cos x 3.4. sin(x + y) = sin x + sin y
9 4. Use the triangles below to complete the following (do not use your calculator, just simplify your answers as far as possible): 4.1. cos sin tan cos sin tan cos sin tan Copy down the following table and fill in your answers from (4) above and use it to answer the questions that follow: cos θ sin θ tan θ Use your table to calculate the following without using a calculator: sin 45 cos cos 60 + tan sin 60 cos Use your table to show that the following are true: sin 60 cos 60 = tan sin cos 2 45 = cos 30 = 1 sin 2 30
10 6. In the following triangles, find the value of the sides marked with letters:
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