Trigonometric Ratios Part 1

Figure 1

We have come across right angled triangles before, when learning about Pythagoras Theorem. Pythagoras theorem shows us to find the length of a side of a right angled triangle, given we have the lengths of the other two sides. THIS MEANS TO USE THE THEOREM, WE NEED TO KNOW THE LENGTHS OF TWO SIDES OF A RIGHT ANGLED TRIANGLE. However, what do we do if we do not know the lengths of two sides, and we need to work out the length of the last side of the right angled triangle. Then we can use trigonometric ratios.
We know that the longest side of a right angled triangle is called the hypotenuse. When in exams, school or anywhere else, we need to find the length of right angled triangle's side and the only information we have is one side's length and an angle (in figure 1 we have labelled the angle as x), then we label the side opposite to the angle as the opposite. Given we have labelled two sides, the last side of the right angled triangle is called the adjacent. When we say label, what we actually mean is that these three names (hypotenuse, opposite and adjacent) represent the lengths of the triangle. So lets apply the trigonometric ratios below.
This page looks at applying the trigonometric ratios to find the lengths of the sides of right angled triangles.
A right angled triangle, like any triangle, has three angles, however one of the angles is a right angle. So you know that when you see or read about a right angled triangle, one out of the three right angles would be a right angle. Given we know one out of the other two angles, and one length of any side of the right angle, we can apply the trigonometric ratios to find the lengths of the other sides.
There are three trigonometric ratios, where x is 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 value of a to 1 decimal place.

Question 1: In figure 2, we need to find the value of a. First we need to label the sides. Obviously, the side opposite to the angle 55° would be labelled as the opposite, the longest side is labelled as the hypotenuse (remember the longest side is always the diagonal side of the right angled triangle), and lastly, the last side is called the adjacent.
Now lets think, we have three ratios. Which one are we going to use? Lets think about it logically. IT IS NOT HARD.
We have been given the length of the hypotenuse (7cm), we have been given the 55° angle and we need to find the length of the opposite side. How do we know that we need to find the length of the opposite side? Well we can see that we need to find the value of a which represents the length of the side which is opposite to the 55° angle.
Which rule out of the three can we use to attain the length of the opposite side?
We can use the sin(x) = Length of the opposite/Length of the hypotenuse rule.
Lets substitute what we know into the equation.
sin(55) = length of the opposite/7
Now we can rearrange to make length of the opposite the subject.
(Multiply both sides by 7)
sin(55) x 7 = (length of the opposite/7) x 7
sin(55) x 7 = length of the opposite
Length of Opposite = sin(55) x 7 = 5.7cm (to 1 decimal place)

Figure 3

Question 2: Find the value of a to 1 decimal place.

Question 2: In figure 3, we need to find the value of a. First we need to label the sides. The diagonal side of the triangle would be labelled as the hypotenuse. The side opposite to the 77° angle would be the opposite side, and the last side will be labelled as the adjacent. Thus, we need to find the length of the adjacent, which is represented in the question with the letter a.
Now lets think about which of the three ratios we would need to use. IT IS NOT HARD. WE JUST HAVE TO LOOK AT THE INFORMATION GIVEN IN THE QUESTION.
We need to find the length of the adjacent, we have the length of the hypotenuse (8cm) and we know that one of the angles in the right angled triangle is 77°.
So we can use the cos(x) = Length of Adjacent/Length of Hypotenuserule.
Lets substitute what we know into the equation.
cos(77) = Length of the Adjacent/8
We can now solve to find the Length of the Adjacent.
cos(77) = Length of the Adjacent/8
(Multiply both sides by 8)
cos(77) x 8 = (Length of the Adjacent/8) x 8
Length of the Adjacent = cos(77) x 8 = 1.8cm (to 1 decimal place)

Figure 4

Question 3: Find the value of a to one decimal place.

Question 3: In figure 4, we can see that we need to find the value of a. First we need to label the sides of the right angled triangle. The longest side which is the hypotenuse is the diagonal side. The opposite side is the side which is opposite to the angle 65°; the length of the opposite side is represented using the letter a. So now we know that we need to find the length of the opposite side. The last side that we need to label is called the adjacent.
Now lets think about which of the three rules we need to use. IT IS NOT HARD. WE JUST HAVE TO LOOK AT THE INFORMATION GIVEN IN THE QUESTION.
We need to find the length of the opposite side, we have the length of the adjacent (4cm) and we have been the angle 65°.
So we can use the tan(x) = Length of Opposite/Length of Adjacent rule.
Lets substitute what we know into the equation.
tan(65) = Length of Opposite/4
We can now solve to find the length of the opposite.
tan(65) = Length of Opposite/4
(Multiply 4 by both sides)
4 x tan(65) = (Length of Opposite/4) x 4
4 x tan(65) = Length of Opposite
Length of Opposite = 4 x tan(65) = 8.6cm (to 1 decimal place)

So what you need to understand is that trigonometric ratios can only be applied to the right angled triangles. Based on what information you have been given in the question, you can use the right rule to find the length of a side.
Trigonometric Ratios part 2 discusses how we can use the ratios to find the size of the angles in a right angled triangle.

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