Addition Formulae

Example 1:

Given that , express in terms of .

Step 1: Expand and using the addition formulae.

Expanding the brackets,

Step 2: Divide each term by and then rearrange. We want to do this because we know that .

We now put all terms on one side because the question wants in terms of .

Factorise from the LHS,

Then divide both sides by ,

Example 2:

Using a suitable expansion,

a)show that .

Single Step: Rewrite as a sum of angles with known exact trigonometric ratios and expand using the addition formula.

can be written as . Therefore:

b)find .

Single Step: Rewrite as angles that we know the exact trigonometric ratios and expand using addition formula.

can be written as . Therefore:

We now rationalise the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which will be .

Expanding the numerator and denominator,

Example 3:
Solve for , giving your answers to significant figures,

a)

Step 1: Recognise the left side of the equation as the addition formula for sine and rewrite it as a single trigonometric expression.

Simplifying the angle and dividing both sides by ,

Step 2: Solve the equation with the modified range.

Modified range:

Note that (this is to check whether the values we find lie in the modified range or not). Find the PV. Ensure you don’t divide by until you’ve found all values within the modified range.

(PV)

Find the SV by working out PV.

PV (SV)

Both the PV and SV are in the modified range. Take the PV and SV and add to both to find the rest of the values in the modified range:

PV

SV

If we continue adding , we will find values larger than and will therefore be out of the modified range. So, we can stop here.

Note: To not lose accuracy, store all these values in the calculator.

Step 3: Un-modify the range to find . Do this by dividing every value by .

b)

Step 1: Recognise the left side of the equation as the addition formula for tan and rewrite it as a single trigonometric expression.

Step 2: Solve the equation with the modified range.

Modified range:

Note that and

Find the PV.

(PV)

Find the SV by working out PV.

PV (SV)

Both the PV and SV are in the modified range. Take the PV and SV and add to both to find the rest of the values in the modified range.

PV

SV

If we continue adding , we will find values larger than and will therefore be out of the modified range. So, we can stop here.

Note: To not lose accuracy, store all these values in the calculator.

Step 3: Un-modify the range to find . Do this by subtracting from every value and then dividing by .