Trigonometry
Basic Functions
As we know, the three basic trigonometric functions are as follows:
sin 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cos 𝜃 =
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
Working:
1. sin 𝛼
As we know, the sine of an angle is the opposite side over the hypotenuse, therefore:
sin 𝛼 =
6.5
sin 𝛼 = 9.7
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Here we plug the values from the triangle into our formula
sin 𝛼 = 0.670103 …
We now compute the answer
sin 𝛼 = 0.67
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
7.2
cos 𝛼 = 9.7
Again, we plug the values in,
cos 𝛼 = 0.742268 …
Compute the answer
cos 𝛼 = 0.74
And round it off
3. tan 𝛼
The tangent ratio of an angle is the opposite side over the adjacent side. Therefore:
tan 𝛼 =
6.5
tan 𝛼 = 7.2
tan 𝛼 = 0.902777 …
tan 𝛼 = 0.90
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
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.
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 = 190.047286 …
And rounded as instructed
𝑥 = 190.0473
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)
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 =
sec 𝜃 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
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
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:
1
cosec 𝜃 =
sin 𝜃
We are looking at angle 𝛼:
1
cosec 𝛼 =
sin 𝛼
The sine of angle 𝛼 is opposite over hypotenuse:
1
cosec 𝛼 =
6.5
9.7
Which simplifies as below (note that this is the same as hypotenuse over opposite):
9.7
cosec 𝛼 =
6.5
Which we can calculate as equal to:
cosec 𝛼 = 1.492307 …
And round as instructed:
cosec 𝛼 = 1.492
2. sec 𝛼
The secant of an angle can be calculated in the following way:
The secant is the reciprocal of the cosine:
1
sec 𝜃 =
cos 𝜃
We are looking at angle 𝛼:
1
sec 𝛼 =
cos 𝛼
The cosine of angle 𝛼 is adjacent over hypotenuse:
1
sec 𝛼 =
7.2
9.7
Which simplifies as below (note that this is the same as hypotenuse over adjacent):
9.7
sec 𝛼 =
7.2
Which we can calculate as equal to:
sec 𝛼 = 1.347222 …
And round as instructed:
sec 𝛼 = 1.347
3. cot 𝛽
The cotangent of an angle can be calculated in the following way:
The cotangent is the reciprocal of the tangent:
1
cot 𝜃 =
tan 𝜃
We are looking at angle 𝛽:
1
cot 𝛽 =
tan 𝛽
The tangent of angle 𝛽 is opposite over adjacent:
1
cot 𝛽 =
7.2
6.5
Which simplifies as below (note that this is the same as adjacent over opposite):
6.5
cot 𝛽 =
7.2
Which we can calculate as equal to:
cot 𝛽 = 0.902777 …
And round as instructed:
cot 𝛽 = 0.903
Exercise:
1. Use your calculator to calculate the following correct to 2 decimal places:
1.1. cos 48°
1.2. 2 sin 35°
1.3. tan2 81°
1.4. 3 sin2 72°
1
1.5. 4 cos 27°
2. 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:
2.1
2.2
2.4
2.5
2.3
2.6
3. 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
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 30°
4.2. sin 30°
4.3. tan 30°
4.4. cos 60°
4.5. sin 60°
4.6. tan 60°
4.7. cos 45°
4.8. sin 45°
4.9. tan 45°
5. Copy down the following table and fill in your answers from (4) above and use it to
answer the questions that follow:
30°
45°
60°
cos 𝜃
sin 𝜃
tan 𝜃
5.1. Use your table to calculate the following without using a calculator:
5.1.1. sin 45° × cos 45°
5.1.2. cos 60° + tan 45°
5.1.3. sin 60° − cos 60°
5.2. Use your table to show that the following are true:
5.2.1.
sin 60°
cos 60°
2
= tan 60°
5.2.2. sin 45° + cos 2 45° = 1
5.2.3. cos 30° = √1 − sin2 30°
6. In the following triangles, find the value of the sides marked with letters:
6.1
6.3
6.5
6.2
6.4
6.6
6.7
6.8