# Trig Formulas

### Trig factor formulas

Trig factor formula is used to convert the product of 2 trigonometric functions into a sum or difference of 2 trigonometric functions or vice-versa.

Sum or Difference of trig functions (involves cosine and sine)

Remember these trig formulas split a complex trigonometry function into a difference/addition of two trigonometry functions.

The trig formulas are derived from the additional formulas.

1)

2)

3)

4)

Combining Trigonometric functions (involves cosine and sine)

We can derive the following trig factor formulas that combine basic trigonometry functions into one. It is a good idea if you are able to memorize the trig formulas.

By placing P=A+B and Q=A-B in 1) to 4) above equations, we obtain

1)

2)

3)

4)

Example

Let us apply the trig formulas to the following problems.

a)Show that

We combined sin ? with sin 3 ? and sin 5 ? with sin7 with the trig formulas.

Left Hand Side =

Take out cos ?. Notice that there are two basic trigonometry functions left in the brackets .We combine them together as well.

=

=

Look at the RHS (Right Hand Side) of the equation. It is in terms of ? and not 4 ?. So we use double angle formula to reduce sin 4 ? into 2sin 2 ? cos ?.

=

Notice that there are cos and sin on the RHS. What we are missing is 2sin cos ?. So we use the double angle formula again to reduce sin 2 into 2sincos

= ( RHS Shown )

Show that

Upon reading the question, we notice that there is only one trigonometry function in the numerator and denominator on the RHS( Right Hand Side) of the equation.

LHS =

=

We have to think on how we are going to get a tangent function on the RHS.

“Could it be ”. We arrange the equation in that manner.

=

=

= ( RHS Proven)

b) Show that

LHS =

We place it in the form 2sin A sin B .Then we expand it using the trigonometry factor formulas.

=

=

=

We note that the power on the right hand side of the equation is 1. The power

of the current equation is at 2. We reduce it using the double angle formula

(cos 2 = 2 cos -1)

=

= ( RHS Shown)