Trigonometry: Finding angles and sides
In a nutshell
sin(), cos(), tan() and their corresponding ratios can be used to find missing lengths and angles in a right-angled triangle.
Calculating a side
PROCEDURE
1. | Label the sides and angle of the triangle. |
2. | Work out what trigonometric ratio you have to use (SOH, CAH or TOA). |
3. | Write down and rearrange the formula to make the missing side the subject. |
4. | Substitute in the values for the other side and the angle to get the length. |
Example 1
Find the value of the side labelled x in the diagram given to 1 decimal place.
Label the sides and angle:
O and H are the given sides. So, use SOH:
sin(50)=HO
So:
O=H×sin(50)
Substitute in the values:
O=10×sin(50)=7.660444....
Round to 1 decimal place, like the question asks:
O=7.7m (1 d.p.)
Calculating an angle
Calculating the missing angle of a right-angled triangle is very similar to calculating the missing length. However, it is important to first learn about inverse trigonometric functions.
The inverse trigonometric functions
Inverse trigonometric functions invert the trigonometric function, giving the value of the angle x.
Function | What it's called | Examples |
sin−1() | Inverse sine | If sin(x)=0.5, then x=sin−1(0.5) |
cos−1() | Inverse cosine | If cos(x)=0.8, then x=cos−1(0.8) |
tan−1() | Inverse tangent | If tan(x)=0.42, then x=tan−1(0.42) |
Note: The inverse function sin−1() is NOT the same as sin()1! It is just notation to tell us that it is the inverse of the sin() function. The same goes for the other two trigonometric functions.
Calculating the missing angle
The process of finding the missing angle of a right-angled triangle is very similar to finding the missing side.
PROCEDURE
1. | Label the sides and angle of the triangle. |
2. | Work out what trigonometric ratio you have to use (SOH, CAH or TOA). |
3. | Write down the corresponding formula. |
4. | Substitute in the values for the sides. |
5. | Perform the appropriate inverse trigonometric function to get the angle. |
Example 2
Find the value of x in the missing diagram to 1 decimal place.
First, label the diagram:
O and A are the important sides here. Hence, use TOA:
tan(x)=adjacentopposite=AO
Substitute in the values:
tan(x)=125
Use the corresponding inverse trigonometric function:
If tan(x)=125 then x=tan−1(125)
x=22.61986....
Round to 1 decimal place, like the question asks:
x=22.6° (1 d.p.)