Half-Angle and Double-Angle Formulas

What are the addition formulas?

In an earlier section, we mentioned one of the addition formulas. The addition formulas are another group of identities that are useful tools for trigonometry. If you memorize these formulas, you can use information from the standard points you learned earlier to find uncommon points without using a calculator.

The sine and cosine addition formulas are:

Using these sine and cosine formulas, you can easily derive the tangent, cotangent, secant, and cosecant addition formulas. Together, we will derive the formula for   in order to get started in our effort to derive the other addition formulas. Pay close attention, because you will be asked to derive one of the addition formulas not given in this section at the end of this module.

Using the addition formulas above, what is the formula for ?

So the addition formula for tangent is .

Use the sine and cosine addition formulas to prove that . The method is the same as the one above.

Challenge : Derive the cotangent, secant, and cosecant addition formulas that have not been given above.

What are the double-angle formulas?

In addition to addition formulas, double-angle formulas are also useful when you are trying to use information from the standard point to find uncommon points without using a calculator. The basic double-angle formulas are:

There are other double-angle formulas, but these three are the most important, and the only double angle formulas that we will cover at this time.

Can you prove that  using the addition formulas?

Let b = a, and use substitution to get

.

Now pull out the sin(a).

Can you prove the double-angle formula   ?

Let b = a, and use substitution to get

Since

,

.

Use substitution to get

Using , , use substitution to get

Can you prove the double-angle formula ?

Now let b = a to get

What are half-angle formulas?

Half-angle formulas are more common than any other type of identity used in trigonometry. Yes, more identities! All of the identities we have learned, including the half-angle formulas, are still not the entirety of the trigonometric identities. The more identities you know, the easier it will be for you to recognize the simplified solution to a complex expression. All of these identities can be used to simplify more complex equations, and can be used to solve for angles using angles that you already know.

The half-angle formulas are:

As you have seen, you can use other identities to derive new ones. The same is true for the half-angle formulas. The rest of this section will cover methods of deriving these formulas, but try and do them yourself before moving ahead. Try using one of the identities that you learned in this module to derive these half-angle formulas.

Did you try to derive the half-angle formulas yourself?

In order to prove that , use the double-angle formula

You can rearrange this formula to get

Now let , and use substitution to get

Divide both sides by 2.

Now, take the square root of both sides.

In order to prove that , use the double-angle formula.

If you did not prove these identities before, stop now and try to prove this one before moving ahead. This method is exactly the same as the one above.

Let . Use substitution.

Add 1 to both sides.

Divide both sides by 2.

Now take the square root of both sides.

Now, prove the half-angle formula  using the sine and cosine half-angle formulas.

Review of New Vocabulary and Concepts

The addition formulas are:

The basic double-angle formulas are:

The half-angle formulas are: