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Double Angle Formula – Explanation and Examples

The double angle formula gives the trigonometric ratio for an angle twice a given angle.

There are double angle formulas for sine and cosine. The formulas for the other trig functions follow from these.

Since the double angle formula gives exact values for trig ratios of minor angles, it is useful for ensuring accuracy in engineering, astronomy, and other physical sciences.

Make sure to understand trig identities fully before reading this article.

This section covers:

  • What is the Double Angle Formula?
  • Proof of the Double Angle Formula
  • Cos Double Angle Formula
  • Sin Double Angle Formula
  • Tangent Double Angle Formula

What is the Double Angle Formula?

The double angle formula is an equation that gives the trigonometric ratio for an angle equal to twice a given angle.

For a trigonometric function $f(x)$, $f(2x) \neq f(x)$. This means that doubling the trig ratio for an angle is not the same as finding the trig ratio for twice the angle. 

Instead, the double angle for $sin(2x)$ is:

$2sinxcosx$.

For cosine, the double angle formula is:

$cos^2x-sin^2x$.

Alternatively, the double angle formula for cosine is written as:

$1-2sin^2x$ or $2cos^2x-1$.

Proof of the Double Angle Formula

The proofs for the double angle formulas come from the sum formulas. 

Recall that the sum formula for $sin(x+y) is $sinxcosy+sinycosx$.

Since $2x=x+x$, $sin(2x) = sinxcosx+sinxcosx$ by the sum formula. This simplifies to $2sinxcosx$.

Now, recall that the sum formula for $cos(x+y)$ is $cosxcosy-sinxsiny$. Therefore, according to the sum formula, $cos(2x) = cosxcosx-sinysiny$. This simplifies to $cos^2x-sin^2x$. 

Then, because $cos^2x = 1-sin^2x$ by the Pythagorean identity, this formula can be written as $1-2sin^2x$. Similarly, because $sin^2x=1-cos^2x$, another way to write the formula is $2cos^2x-1$.

Cos Double Angle Formula

There are actually three double angle formulas for cosine. These all come from the sum formula and are different ways of writing the same expression.

The double angle formulas for cosine are:

  • $cos^2x-sin^2x$
  • $1-2sin^2x$
  • $2cos^2x-1$

Sin Double Angle Formula

Unlike cosine, there is only one double angle formula for sine. Like cosine, however, both the sine and cosine ratios of the original angle must be known for the formula.

The double angle formula for sine is:

$2sinxcosx$.

Tangent Double Angle Formula

It is possible to write the double angle formulas for the other trigonometric functions in terms of sine and cosine. For example, the tangent double angle formula is:

$\frac{2tanx}{1-tan^2x}$.

The proof for this formula is in example 1 below.

Examples

This section goes over common examples of problems involving the double angle formula and their step-by-step solutions.

Example 1

Use the double angle formulas for sine and cosine to prove that the double angle formula for tangent is $\frac{2tanx}{1-tan^2x}$.

Solution

Recall that $tan(2x) = \frac{sin(2x)}{cos(2x)}$.

Then, replace $sin(2x)$ and $cos(2x)$ with their respective double angle formulas to get:

$\frac{2sinxcosx}{cos^2x-sin^2x}$.

In this case, there are multiple ways to simplify. It helps to look at the original formula and remember there is a tangent in the numerator. Since $tanx=\frac{sinx}{cosx}$, it may help to have sine in the numerator of this fraction as well. This means it may help to change the cosine double angle formula from $cos^2x-sin^2x$ to $2cos^2x-1$ since it only has one trig function.

$\frac{2sinxcosx}{2cos^2x-1}$.

Now, factor the function $cos^2x$ from both terms in the denominator:

$\frac{2sinxcosx}{cos^2x(2-sec^2x)}$.

Now, one of the cosines from the denominator will cancel the cosine in the numerator. The other will make the $sinx$ in the numerator into $tanx$. Now, the formula is:

$\frac{2tanx}{2-sec^2x}$.

Finally, recall that $sec^2x-1 = tan^2x$. Since the denominator can be rewritten $-(sec^2x-1-1)$, this simplifies to:

$\frac{2tanx}{-(tan^2x-1)}=\frac{2tanx}{1-tan^2x}$.

Example 2

What is the double angle formula for secant?

Solution

Use the double angle formula for cosine to derive the double angle formula for secant.

Recall that $secx=\frac{1}{cosx}$. Thus, the double angle formula for secant will be:

$\frac{1}{cos(2x)}$.

Now, plug in the formula for $cos(2x)$ to get:

$\frac{1}{cos^2x-sin^2x}$.

This is a formula, but one that uses secant only might be better. Divide both terms in the denominator by $cos^2x$ to get:

$\frac{1}{cos^2x(1-tan^2x)}$

Now, note that denominator can be rewritten as $-(tan^2x-1+1-1)$. Since  $tan^2x+1 = sec^2x$, this simplifies as:

$\frac{sec^2x}{-(sec^2x-2} = \frac{sec^2x}{2-sec^2x}$.

Example 3

Given that sine of an angle $\theta$ is $1-\sqrt{2}$, use the Pythagorean identity and the double angle formula to find $sin(2\theta)$. Assume a positive value for cosine.

Solution

The formula for the sine of a double angle is $sin(2x)=2sinxcosx$. Only the sine value for the angle was given, however, so it is necessary to find the cosine ratio for the original angle. The Pythagorean identity $cos^2x+sin^2x=1$ provides a way.

$cos^2x+(1-\sqrt{2})^2=1$.

$cos^2x=1-2\sqrt{2}+2 = 3-2\sqrt{2}$.

Then, taking the square root of both sides yields:

$cosx=\sqrt{3-2\sqrt{2}}$.

Thus, plugging these values into the formula for the sine of a double angle is:

$2(\sqrt{3-2\sqrt{2}}(1-\sqrt{2})$.

This works fine as an answer, though the whole expression simplifies nicely to:

$-2(\sqrt{2}-1)^2$.

Example 4

Given that cosine of an angle $\theta$ is $1-\sqrt{3}$, use the double angle formula to find $cos(2\theta)$.

Solution

Only the cosine value of the original angle was given. There is a formula for the cosine of a double angle that uses only cosine values. To avoid using the Pythagorean Identity to find sine, use this formula.

Plugging in the given cosine ratio yields:

$2(1-\sqrt{3})^2-1$.

Simplifying yields:

$2(1-2\sqrt{3}+3)-1 = 2(4-2\sqrt{3})-1 = 8-4sqrt{3}-1 = 7-4sqrt{3}$.

Example 5

Given that sine of an angle $\theta$ is $2-\sqrt{2}$, find $csc(2\theta)$.

Solution

Since cosecant is the reciprocal of sine, find $sin(2\theta)$ and its reciprocal.

The double angle formula for sine, however, requires cosine to be known. Since $cos^2x=1-sin^2x$, the cosine of this angle is $\sqrt{1-(2-\sqrt{2})^2} = \sqrt{1-(4-4\sqrt{2}+2)} = \sqrt{1-(6-4\sqrt{2})} = \sqrt{4\sqrt{2}-5}$.

Using the double angle formula for sine, then, yields:

$sin(2\theta) = 2(2-\sqrt{2})(\sqrt{4\sqrt{2}-5) = (4-2\sqrt{2})\sqrt{4\sqrt{2}-5}$.

Then, the cosecant of $2\theta$ will be the reciprocal of this:

$\frac{1}{(4-2\sqrt{2})\sqrt{4\sqrt{2}-5}}$.

Practice Questions

1. Which of the following is a double angle formula equivalent to cosecant?

2. If the sine of an angle $\theta$ is $3-2\sqrt{2}$, what is the sine of $2\theta$?

3. Which of the following is equivalent to the quadruple angle formula for sine?

4. If the tangent of an angle $\theta$ is $\sqrt{5}$, what is the tangent of $2\theta$?

5. If the cosine of an angle $\theta$ is $5 – 4\sqrt{2}$, what is the cosine of $2\theta$?


 

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