Mathematical Mischief (2012)
The Trigonometric Ratios!
So, triangles...
Who likes them?
Most people do. Then we get to mathematical triangles, and people start to cry. Don’t worry, it’s not
hard! We’ll work through it, and at the end, you’ll be a professional at this. :)
So what is a triangle?
A triangle is a geometric shape that is made up of three straight lines.
They can come in many different forms.
Some look like this:
But there are many others.
Now, let’s say we want to compare the sides of a triangle with the angles that are inside it.
How do we do that?
What we do, is utilize the trigonometric ratios. They are a series of ratios, designed to help us
understand the ratio of angles to sides.
Opposite
Firstly, we take a right angled triangle, and label a triangle, based around a single angle. Here’s an
example.
Hy
po
ten
us
e
x
Adjacent
Now, I’ve already labelled this for you, but there are three important points to note:
• Hypotenuse - This is the longest side of the triangle.
• Adjacent - This is the side adjacent to the angle, that is not the hypotenuse.
• Opposite - This is the remaining angle, the side that’s directly opposite the angle.
We use these values, alongside special trigonometric functions, called the sine, the cosine, and
the tangent, in order to help us understand these ratios.
Let’s begin with the sine.
Josh Young - Mathematical Mischief
Mathematical Mischief (2012)
The Sine Function
The sine function refers to the ratio between the lengths of the opposite and hypotenuse sides of
the triangle. It is calculated as this:
sin(x) =
O
H
Let’s apply this:
Example 1:
Take the following triangle:
5
13
x
12
Define the value of the sine function, in terms of the side lengths of the triangle.
Step 1. State the sine function, and the values of the opposite and the hypotenuse.
O
sin(x) =
H
O=5
H = 13
Step 2. State the value of the sine function by substituting
the values of the opposite and hypotenuse.
O
sin(x) =
H
5
sin(x) =
13
We can use this function in order to find the angle of the triangle, by taking the inverse sine of the
opposite and hypotenuse. Note: The angle will always be more than zero, but less than 180
degrees.
The inverse sine function, used to determine unknown angles in these cases, is calculated with the
formula:
⎛ O⎞
sin −1 ⎜ ⎟ = x
⎝H⎠
Let’s apply the values from our original example to find the unknown angle x.
Josh Young - Mathematical Mischief
Mathematical Mischief (2012)
Example 2: Use the sine value determined from Example 1 to determine the unknown angle.
Step 1. State the inverse sine function, and the values of the opposite and the
hypotenuse.
⎛ O⎞
sin −1 ⎜ ⎟ = x
⎝H⎠
O=5
H = 13
Step 2. State the value of the angle by substituting
the values of the opposite and hypotenuse.
⎛ O⎞
sin −1 ⎜ ⎟ = x
⎝H⎠
⎛ 5⎞
sin −1 ⎜ ⎟ = x
⎝ 13 ⎠
⎛ 5⎞
sin −1 ⎜ ⎟ = 23.62°
⎝ 13 ⎠
The Cosine Function
The cosine function refers to the ratio between the lengths of the adjacent and hypotenuse sides of
the triangle. It is calculated as this:
cos(x) =
A
H
Example 3:
Take the following triangle:
5
2
2
x
4
Define the value of the cosine function, in terms of the side lengths of the triangle.
Josh Young - Mathematical Mischief
Mathematical Mischief (2012)
Step 1. State the cosine function, and the values of the adjacent and the hypotenuse.
A
cos(x) =
H
A=4
H =2 5
Step 2. State the value of the cosine function by substituting
the values of the adjacent and hypotenuse.
A
H
4
cos(x) =
2 5
2
cos(x) =
5
cos(x) =
We can use this function in order to find the unknown angle in the triangle, by taking the inverse
cosine of the adjacent and hypotenuse. Note: The angle will always be more than zero, but less
than 180 degrees.
The inverse cosine function, used to determine unknown angles in these cases, is calculated with
the formula:
⎛ A⎞
cos −1 ⎜ ⎟ = x
⎝H⎠
Let’s apply the values from our original example to find the unknown angle x.
Example 4: Use the cosine value determined from Example 3 to determine the unknown angle.
Step 1. State the inverse cosine function, and the values of the adjacent and the
hypotenuse.
⎛ A⎞
cos −1 ⎜ ⎟ = x
⎝H⎠
A=4
H =2 5
Step 2. State the value of the angle by substituting
the values of the adjacent and hypotenuse.
⎛ A⎞
cos −1 ⎜ ⎟ = x
⎝H⎠
⎛ 2 ⎞
cos −1 ⎜
=x
⎝ 5 ⎟⎠
⎛ 2 ⎞
cos −1 ⎜
= 26.57°
⎝ 5 ⎟⎠
Josh Young - Mathematical Mischief
Mathematical Mischief (2012)
The Tangent Function
The tangent function refers to the ratio between the lengths of the opposite and adjacent sides of
the triangle. It is calculated as this:
tan(x) =
O
A
Example 5:
Take the following triangle:
5
5
5
x
10
Define the value of the tangent function, in terms of the side lengths of the triangle.
Step 1. State the tangent function, and the values of the adjacent and the opposite.
O
tan(x) =
A
O=5
A = 10
Step 2. State the value of the tangent function by substituting
the values of the adjacent and opposite.
O
tan(x) =
A
5
tan(x) =
10
1
tan(x) =
2
We can also use this function in order to find the unknown angle of the triangle, x, by taking the
inverse tangent of the opposite and adjacent. Note: The angle will always be more than zero, but
less than 180 degrees. The inverse tangent is calculated using the formula:
⎛ O⎞
tan −1 ⎜ ⎟ = x
⎝ A⎠
Josh Young - Mathematical Mischief