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Figure 1
Trigonometric ratios can be used not only to find the length of a side, but also the size of an angle within the right angled triangle. Before going through this page, please make sure that you understand how to label each side of the right angled triangle, as discussed in the Trigonometric Ratios Part 1 page. Remember we are using the same trigonometric ratios (we can apply them to only right angled triangles) that we used in Trigonometric Ratios Part 1 page.
Remember that a right angled triangle has three angles, one is a right angle (90 degrees), as shown in figure 1, we have labelled the other two angles x and y. You can choose what letters you want to use to represent the size of the angles; we have chosen x and y. In the questions below, the angle we need to find is represented using the letter a.
To find the size of an angle, which we do not know, we need to work with the lengths of the right angled triangle. Then depending upon what type of side we have the lengths for (hypotenuse, opposite and adjacent), we can use the appropriate trigonometric ratio.
You should know the trigonometric ratios from the Part 1 page; they are written below.
x represents the size of the angle.
Remember that a right angled triangle has three angles, one is a right angle (90 degrees), as shown in figure 1, we have labelled the other two angles x and y. You can choose what letters you want to use to represent the size of the angles; we have chosen x and y. In the questions below, the angle we need to find is represented using the letter a.
To find the size of an angle, which we do not know, we need to work with the lengths of the right angled triangle. Then depending upon what type of side we have the lengths for (hypotenuse, opposite and adjacent), we can use the appropriate trigonometric ratio.
You should know the trigonometric ratios from the Part 1 page; they are written below.
x represents the size of the angle.
- sin(x) = Length of Opposite/ Length of Hypotenuse
- cos(x) = Length of Adjacent/ Length of Hypotenuse
- tan(x) = Length of Opposite/ Length of Adjacent
Figure 2
Question 1: Find the size of the angle a in figure 2 to 1 decimal place.
To work out question 1, we need to look at figure 2. Lets first label the sides, from the perspective of angle a (the angle we need to find). If you do not know how to label the sides, we have discussed this in the Trigonometric Ratios Part 1 page.
So the side with length 5cm is the opposite side to the angle a. The longest side is the diagonal; this is the hypotenuse. We can see that the length of the hypotenuse is not given in the question. The last side we need to label is the adjacent; the length of the adjacent is given in the question and thus is 7cm.
If we wanted to know the size of the hypotenuse, we can find the length using Pythagoras Theorem.
So
Length of Opposite = 5cm (Given in figure 2)Length of Adjacent = 7cm (Given in figure 2)
Which trigonometric ratio out of the three can we use which contains the above information?
We can use the sin(x) = Length of the Opposite/Length of the Hypotenuse rule.
Lets substitute what we know into the equation. Remember that the 3 trigonometric ratios, have used the letter x to represent the size of angle; our question has used a.
So
sin(a) = Length of the Opposite/Length of the Hypotenuse
sin(a) = 5/7
How do we find the value of a? We take the inverse of the sine function.
If sin(a) = 5/7, then a = sin-1(5/7)
a = sin-1(5/7) = 45.6° (to 1 decimal place)
So the side with length 5cm is the opposite side to the angle a. The longest side is the diagonal; this is the hypotenuse. We can see that the length of the hypotenuse is not given in the question. The last side we need to label is the adjacent; the length of the adjacent is given in the question and thus is 7cm.
If we wanted to know the size of the hypotenuse, we can find the length using Pythagoras Theorem.
So
Length of Opposite = 5cm (Given in figure 2)Length of Adjacent = 7cm (Given in figure 2)
Which trigonometric ratio out of the three can we use which contains the above information?
We can use the sin(x) = Length of the Opposite/Length of the Hypotenuse rule.
Lets substitute what we know into the equation. Remember that the 3 trigonometric ratios, have used the letter x to represent the size of angle; our question has used a.
So
sin(a) = Length of the Opposite/Length of the Hypotenuse
sin(a) = 5/7
How do we find the value of a? We take the inverse of the sine function.
If sin(a) = 5/7, then a = sin-1(5/7)
a = sin-1(5/7) = 45.6° (to 1 decimal place)
Figure 3
Question 2: Find the size of the angle a in figure 3.
To work out question 2, we need to look at figure 3. Lets first label each side of the right angled triangle from the perspective of angle a. The diagonal side is the hypotenuse, as this is the longest side. We have been given the length of the hypotenuse in figure 2, it is 10cm. The opposite side to angle a is obviously called the opposite side; we have not been given the length of this. The last side is the adjacent and the length of the adjacent is 8cm.
If we wanted to know the length of the opposite side, we can find the length using Pythagoras Theorem.
So
Length of hypotenuse = 10cmLength of adjacent = 8cm
Which trigonometric ratio out of the three can we use which contains the above information?
We can use
cos(x) = Length of Adjacent/ Length of Hypotenuse
Lets substitute what we know into the equation. Remember that the 3 trigonometric ratios have used the letter x to represent the size of angle; our question has used a.
So
cos(a) = Length of Adjacent/ Length of Hypotenuse
cos(a) = 8/10
How do we find the value of a? We take the inverse of the cosine function.
If cos(a) = 8/10, a = cos-1(8/10)
a = cos-1(8/10) = 36.9° (to 1 decimal place)
If we wanted to know the length of the opposite side, we can find the length using Pythagoras Theorem.
So
Length of hypotenuse = 10cmLength of adjacent = 8cm
Which trigonometric ratio out of the three can we use which contains the above information?
We can use
cos(x) = Length of Adjacent/ Length of Hypotenuse
Lets substitute what we know into the equation. Remember that the 3 trigonometric ratios have used the letter x to represent the size of angle; our question has used a.
So
cos(a) = Length of Adjacent/ Length of Hypotenuse
cos(a) = 8/10
How do we find the value of a? We take the inverse of the cosine function.
If cos(a) = 8/10, a = cos-1(8/10)
a = cos-1(8/10) = 36.9° (to 1 decimal place)
To answer question 3, we have to look at figure 4. Lets first label the sides of the right angled triangle. The side with length 8cm is opposite to angle a and thus this is obviously called the opposite side. The diagonal side is the longest side of the right angled triangle and thus this is the hypotenuse. The last side of the right angled triangle is the adjacent. We have been given the length of the adjacent and it is 3cm.
If we wanted to know the length of the hypotenuse, we can find the length using Pythagoras Theorem.
So
Length of Adjacent = 3cmLength of Opposite = 8cm
Which trigonometric ratio out of the three can we use which contains the above information?
We can use
tan(x) = Length of Opposite/ Length of Adjacent
Lets substitute what we know into the equation. Remember that the 3 trigonometric ratios have used the letter x to represent the size of angle; our question has used a.
So
tan(a) = 8/3
How do we find the value of a? We take the inverse of the cosine function.
If tan(a) = 8/3, a = tan-1(8/3)
a = tan-1(8/3) = 69.4° (to 1 decimal place)
If we wanted to know the length of the hypotenuse, we can find the length using Pythagoras Theorem.
So
Length of Adjacent = 3cmLength of Opposite = 8cm
Which trigonometric ratio out of the three can we use which contains the above information?
We can use
tan(x) = Length of Opposite/ Length of Adjacent
Lets substitute what we know into the equation. Remember that the 3 trigonometric ratios have used the letter x to represent the size of angle; our question has used a.
So
tan(a) = 8/3
How do we find the value of a? We take the inverse of the cosine function.
If tan(a) = 8/3, a = tan-1(8/3)
a = tan-1(8/3) = 69.4° (to 1 decimal place)
To be able to use trigonometric ratios, you need to understand how to label each side. Then given how much information you have, you are able to find the size of an angle inside a right angled triangle or the length of the right angled triangle's side.
REMEMBER TRIGONOMETRIC RATIOS CAN ONLY BE APPLIED TO RIGHT ANGLED TRIANGLES.
REMEMBER TRIGONOMETRIC RATIOS CAN ONLY BE APPLIED TO RIGHT ANGLED TRIANGLES.
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