Solving Basic Trigonometric Functions in Radians
Neil Trivedi
Teacher
Solving Trigonometric Equations in Radians
General Steps on How to Solve Trigonometric Equations:
1. Rearrange for either sin, cos, or tan if needed.
2. Modify the range if needed.
3. Apply the inverse function of either sin, cos, or tan. This will give us the primary value (PV).
4. Find the secondary value (SV) by applying the following rules:
PV
PV
PV
5. Add or subtract to find other values that may fall within the range. This is because every trigonometric graph cycles every . To note, for tan, we can add and subtract from the PV to find all the values in the range, but the PV/SV approach is a one fits all method.
6. Un-modify the range if needed.
Example 1:
Solve for ,
Step 1: The current range is for . We need to modify the range to read . To do this, multiply everything by .
Step 2: Apply the inverse function of tan to get the primary value (PV).
(PV)
Step 3: Find the secondary value (SV). For tan, we work out PV.
PV (SV)
Step 4: Both the PV and SV are less than and are, therefore, in the modified range. Take the PV and SV and add to both to find two other values in the modified range.
If we continue adding , we will find values larger than and will therefore be out of the modified range. So, we can stop here.
Step 5: Un-modify the range to find . Do this by dividing every value by .
No answer provided.
Example 2:
Solve for ,
Step 1: Rearrange the equation to make the function (sin) we’re solving for the subject.
Step 2: The current range is for . We need to modify the range to read . To do this, multiply everything by and then add .
Step 3: Apply the inverse function of sin to get the primary value (PV).
(PV)
Step 4: Find the secondary value (SV). For sin, we work out PV.
PV (SV)
Step 5: The SV is in the modified range, but the PV is not. However, we can still use the PV to find other values in the range. Add to both the PV and SV to find the other values in the modified range.
Note: We can stop here but it is worth checking in the real exam that when you continue adding , you find values outside the modified range.
Step 6: Un-modify the range to find . Do this by subtracting and then dividing by .
No answer provided.
Example 3:
Solve for , giving your answers to decimal places when necessary,
Step 1: Suppose and then factorise the equation.
or
Then, replace with .
or
Step 2: The current range is for . We need to modify the range to read . To do this, multiply everything by .
Step 3: For , apply the inverse function of cos to get the primary value (PV).
(PV)
Note: It is worth storing your values in your calculator, so you don’t lose any accuracy.
Step 4: Find the secondary value (SV). For , we work out PV.
PV (SV)
Step 5: Both the PV and SV are in the modified range. We can now subtract from both the PV and SV to find more solutions inside the modified range.
All four angles found are within the modified range.
Step 6: Un-modify the range to find . Do this by dividing every value by .
Step 7: For , apply the inverse function of cos to get the primary value (PV).
(PV)
Step 8: Find the secondary value (SV). For cos, we work out PV.
PV (SV)
Note: Interestingly, the PV SV here. We know this as on the cosine graph, it has a minimum point at where the angle is in the range between and .
Step 9: Subtract from the PV/SV to find one more solution in the range.
Both and are within the modified range.
Step 10: Un-modify the range to find . Do this by dividing every value by .
Therefore, all of our solutions are:
and
No answer provided.
Practice Questions