Trigonometry: Double Angle Formulas

Double angle formulas are not just another set of trigonometric identities to memorize; they are powerful tools for simplifying complex expressions, solving intricate equations, and computing exact values for angles you don't have on the standard unit circle. Mastering these formulas is essential for success in calculus, physics, and engineering, where they are used to model oscillatory behavior, analyze waveforms, and integrate trigonometric functions.

Core Concept: Derivation from Sum Identities

You don't need to memorize the double angle formulas by rote if you understand where they come from. They are a direct application of the sum identities you already know. The sum formulas state:

$$\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$$

$$\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$$

To derive the double angle formulas, you simply let $\alpha =\theta$ and $\beta =\theta$. Substituting into the sum formulas gives you the foundational double angle identities:

For sine: $$\sin (2\theta )=\sin (\theta +\theta )=\sin \theta \cos \theta +\cos \theta \sin \theta =2\sin \theta \cos \theta$$

For cosine: $$\cos (2\theta )=\cos (\theta +\theta )=\cos \theta \cos \theta -\sin \theta \sin \theta ={\cos }^2\theta -{\sin }^2\theta$$

The tangent formula follows similarly from $\tan (\alpha +\beta )$: $$\tan (2\theta )=\frac{2\tan \theta }{1-{\tan }^2\theta }$$

These derivations solidify your understanding, transforming the formulas from arbitrary facts into logical consequences of a more general rule.

The Three Forms of $\cos (2\theta )$

The formula for $\cos (2\theta )$ is particularly versatile because it can be expressed in three different ways using the Pythagorean identity ${\sin }^2\theta +{\cos }^2\theta =1$. This is crucial for simplifying expressions depending on what you are given.

1. $\cos (2\theta )={\cos }^2\theta -{\sin }^2\theta$ (The direct derivation).
2. If you solve the Pythagorean identity for ${\sin }^2\theta =1-{\cos }^2\theta$ and substitute, you get:

$\cos (2\theta )=2{\cos }^2\theta -1$

1. If you solve for ${\cos }^2\theta =1-{\sin }^2\theta$ instead, you get:

$\cos (2\theta )=1-2{\sin }^2\theta$

Choosing the correct form is a key problem-solving skill. For example, if an expression is rich in cosine terms, use form 2. If it's rich in sine terms, use form 3. This choice directly streamlines the simplification process.

Applications: Simplification and Exact Value Evaluation

The primary application of these formulas is to rewrite expressions involving $2\theta$ in terms of $\theta$, which are often simpler. Consider the expression $6\sin \theta \cos \theta$. Using the sine double angle formula in reverse, you can immediately see it simplifies to $3\sin (2\theta )$.

A more powerful application is evaluating exact values for non-standard angles. You know the exact trigonometric values for angles like $30^{\circ }$, $45^{\circ }$, and $60^{\circ }$. The double angle formulas let you find exact values for their doubles, like $15^{\circ }$ or $22.5^{\circ }$, through a reverse process.

Example: Find the exact value of $\sin (22.5^{\circ })$.

1. Note that $22.5^{\circ }$ is half of $45^{\circ }$. So, if $\theta =22.5^{\circ }$, then $2\theta =45^{\circ }$.
2. Use the cosine double angle form that involves sine: $\cos (2\theta )=1-2{\sin }^2\theta$.
3. Substitute $2\theta =45^{\circ }$: $\cos (45^{\circ })=1-2{\sin }^2(22.5^{\circ })$.
4. You know $\cos (45^{\circ })=\frac{\sqrt{2}}{2}$. Solve for $\sin (22.5^{\circ })$:

$$\frac{\sqrt{2}}{2}=1-2{\sin }^2(22.5^{\circ })$$ $$2{\sin }^2(22.5^{\circ })=1-\frac{\sqrt{2}}{2}=\frac{2-\sqrt{2}}{2}$$ $${\sin }^2(22.5^{\circ })=\frac{2-\sqrt{2}}{4}$$ $$\sin (22.5^{\circ })=\sqrt{\frac{2-\sqrt{2}}{4}}=\frac{\sqrt{2-\sqrt{2}}}{2}\text{(positive since 22.5∘ is in QI)}$$

Solving Equations and Proving Identities

Double angle formulas are indispensable for solving trigonometric equations that involve functions of multiple angles. They allow you to rewrite the equation in terms of a single argument ($\theta$), making it solvable.

Example: Solve $\cos (2\theta )+\sin \theta =0$ for $0\le \theta <2\pi$.

1. Use the identity $\cos (2\theta )=1-2{\sin }^2\theta$ to rewrite everything in terms of $\sin \theta$:

$1-2{\sin }^2\theta +\sin \theta =0$

1. Rearrange: $-2{\sin }^2\theta +\sin \theta +1=0$ or $2{\sin }^2\theta -\sin \theta -1=0$.
2. Factor: $(2\sin \theta +1)(\sin \theta -1)=0$.
3. Solve each factor:

• $\sin \theta =1$ $\Rightarrow$ $\theta =\frac{\pi }{2}$
• $\sin \theta =-\frac{1}{2}$ $\Rightarrow$ $\theta =\frac{7\pi }{6},\frac{11\pi }{6}$

When proving other identities, double angle formulas act as a bridge. You often pick one side of the identity (typically the more complex one) and manipulate it using double angle formulas and other identities until it matches the other side. The three forms of $\cos (2\theta )$ give you strategic options for substitution.

Deeper Insight: Power-Reducing Formulas

A direct and immensely useful corollary of the double angle formulas for cosine are the power-reducing formulas. They are derived by solving the $\cos (2\theta )$ identities for ${\sin }^2\theta$ and ${\cos }^2\theta$.

From $\cos (2\theta )=1-2{\sin }^2\theta$, you get: $${\sin }^2\theta =\frac{1-\cos (2\theta )}{2}$$

From $\cos (2\theta )=2{\cos }^2\theta -1$, you get: $${\cos }^2\theta =\frac{1+\cos (2\theta )}{2}$$

These formulas "reduce" even powers of sine and cosine to expressions involving first powers of cosine of a double angle. This is the standard technique in calculus for integrating even powers of sine and cosine, such as $\int {\sin }^2\theta d\theta$.

Common Pitfalls

1. Misapplying the Sign in Cosine Formulas: Remember, $\cos (2\theta )={\cos }^2\theta -{\sin }^2\theta$. A common error is writing ${\cos }^2\theta +{\sin }^2\theta$, which simply equals 1. Always associate the minus sign with the cosine double angle derivation.

1. Forgetting the Alternative Forms: Using ${\cos }^2\theta -{\sin }^2\theta$ in every situation can lead to messy algebra. If you see $1+\cos (2\theta )$ in an expression, recognize it as a signal to use the form $2{\cos }^2\theta$. If you see $1-\cos (2\theta )$, that signals the form $2{\sin }^2\theta$.

1. Overcomplicating Simplification: When simplifying an expression like $4-8{\sin }^2\theta$, recognize it as $4(1-2{\sin }^2\theta )$, which is $4\cos (2\theta )$. Look for these factorable patterns directly related to the double angle forms instead of expanding unnecessarily.

1. Domain Errors with Tangent: The formula $\tan (2\theta )=\frac{2\tan \theta }{1-{\tan }^2\theta }$ is only valid when $\tan \theta$ is defined and ${\tan }^2\theta \ne 1$ (i.e., $\theta \ne 45^{\circ }+90k^{\circ }$). Also, if $\theta$ is such that $2\theta$ is an odd multiple of $90^{\circ }$, $\tan (2\theta )$ is undefined, but the right-hand side may misleadingly yield a number. Always consider the domain of the original functions.

Summary

• Double angle formulas ($\sin 2\theta =2\sin \theta \cos \theta$, $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta$, $\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$) are derived directly from the sum identities by setting both angles equal.
• The cosine formula has three equivalent forms (${\cos }^2\theta -{\sin }^2\theta$, $2{\cos }^2\theta -1$, $1-2{\sin }^2\theta$), and choosing the right one is key to efficient simplification and equation solving.
• These formulas enable the evaluation of exact trigonometric values for non-standard angles (like $15^{\circ }$ or $22.5^{\circ }$) by relating them to known values of their double or half.
• They are essential tools for solving trigonometric equations and proving more complex identities by rewriting expressions in terms of a single angle.
• Solving the cosine formulas for ${\sin }^2\theta$ and ${\cos }^2\theta$ yields the power-reducing formulas, which are critical for calculus operations involving even powers of sine and cosine.