9.3 Double-Angle, Half-Angle, and Reduction Formulas

Double-Angle Formulas

Double-angle formulas let you express trig functions of $2\theta$ using functions of the original angle $\theta$. They show up constantly when simplifying expressions and solving trig equations, so knowing them cold will save you a lot of time.

Core Double-Angle Formulas

Sine double-angle:

$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$

Cosine double-angle (three equivalent forms):

$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$

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

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

The cosine formula has three versions because you can substitute $\sin^2(\theta) = 1 - \cos^2(\theta)$ or $\cos^2(\theta) = 1 - \sin^2(\theta)$ into the first form. Which version you pick depends on what's in your problem. If you only know $\cos(\theta)$, use $2\cos^2(\theta) - 1$. If you only know $\sin(\theta)$, use $1 - 2\sin^2(\theta)$.

Tangent double-angle:

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

Solving Equations with Double-Angle Formulas

When you see an equation like $\cos(2\theta) = \frac{1}{2}$, here's how to approach it:

1. Treat $2\theta$ as a single variable. Find all angles where cosine equals $\frac{1}{2}$: that's $2\theta = \frac{\pi}{3}$ and $2\theta = \frac{5\pi}{3}$ (plus full rotations).
2. Add the general period: $2\theta = \frac{\pi}{3} + 2\pi k$ and $2\theta = \frac{5\pi}{3} + 2\pi k$, where $k$ is any integer.
3. Divide everything by 2: $\theta = \frac{\pi}{6} + \pi k$ and $\theta = \frac{5\pi}{6} + \pi k$.

Alternatively, you might need to replace $\cos(2\theta)$ with one of its expanded forms to get an equation entirely in terms of $\sin(\theta)$ or $\cos(\theta)$, then solve from there.

Reduction Formulas and Half-Angle Formulas

Reduction Formulas

Reduction formulas let you rewrite trig functions of angles outside the first quadrant in terms of first-quadrant angles. They're really just specific applications of the sum/difference identities you already know.

Shifting by $\pi$ (180°):

• $\sin(\theta + \pi) = -\sin(\theta)$
• $\cos(\theta + \pi) = -\cos(\theta)$
• $\tan(\theta + \pi) = \tan(\theta)$

Adding $\pi$ puts you in the opposite quadrant, which flips the sign of sine and cosine. Tangent stays the same because both sine and cosine flip, and a negative divided by a negative is positive.

Shifting by $\frac{\pi}{2}$ (90°):

• $\sin\!\left(\theta + \frac{\pi}{2}\right) = \cos(\theta)$
• $\cos\!\left(\theta + \frac{\pi}{2}\right) = -\sin(\theta)$
• $\tan\!\left(\theta + \frac{\pi}{2}\right) = -\cot(\theta)$

Be careful with the $\pm$ signs here. The signs depend on whether you're adding or subtracting $\frac{\pi}{2}$, so don't just memorize a single version. Think about which quadrant the resulting angle lands in.

Example: To find $\sin\!\left(\frac{5\pi}{4}\right)$, recognize that $\frac{5\pi}{4} = \frac{\pi}{4} + \pi$. Applying the reduction formula: $\sin\!\left(\frac{\pi}{4} + \pi\right) = -\sin\!\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$.

Half-Angle Formulas

Half-angle formulas express trig functions of $\frac{\theta}{2}$ using the original angle $\theta$. They're derived directly from the cosine double-angle formulas (solved for $\sin$ or $\cos$).

$\sin\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$

$\cos\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$

$\tan\!\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} = \frac{\sin(\theta)}{1 + \cos(\theta)}$

The $\pm$ on sine and cosine is not optional decoration. You choose positive or negative based on which quadrant $\frac{\theta}{2}$ falls in, not $\theta$ itself. The tangent forms don't need the $\pm$ because the sign is already determined by the signs of the numerator and denominator.

Choosing the correct sign:

If $\theta$ is in Quadrant II (between $\frac{\pi}{2}$ and $\pi$), then $\frac{\theta}{2}$ is in Quadrant I (between $\frac{\pi}{4}$ and $\frac{\pi}{2}$). In Quadrant I, both sine and cosine are positive, so you'd use the positive root for both.

Example: Find the exact value of $\cos\!\left(\frac{\pi}{8}\right)$.

1. Recognize that $\frac{\pi}{8} = \frac{1}{2} \cdot \frac{\pi}{4}$, so set $\theta = \frac{\pi}{4}$.
2. Apply the half-angle formula: $\cos\!\left(\frac{\pi}{8}\right) = \pm\sqrt{\frac{1 + \cos\!\left(\frac{\pi}{4}\right)}{2}}$
3. Substitute $\cos\!\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$: $\cos\!\left(\frac{\pi}{8}\right) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \frac{\sqrt{2 + \sqrt{2}}}{2}$
4. Since $\frac{\pi}{8}$ is in Quadrant I, cosine is positive, so you take the positive root.

Verifying Identities

When a problem asks you to verify an identity using these formulas, the strategy is:

1. Pick the more complicated side of the equation to work with.
2. Substitute the appropriate double-angle, half-angle, or reduction formula.
3. Simplify using algebra (factor, combine fractions, use Pythagorean identities).
4. Show that it equals the other side.

The key skill is recognizing which formula to apply. If you see $2\theta$, think double-angle. If you see $\frac{\theta}{2}$, think half-angle. If you see an angle like $\theta + \pi$, think reduction.

Fundamental Concepts in Trigonometry

The unit circle and angle measurement

• Radians and degrees are two ways to measure angles. A full rotation is $360°$ or $2\pi$ radians.
• The unit circle (radius = 1, centered at the origin) defines trig function values: for any angle $\theta$, the point on the circle is $(\cos\theta, \sin\theta)$.
• The quadrant determines the sign of each function. Sine is positive in Quadrants I and II; cosine is positive in Quadrants I and IV; tangent is positive in Quadrants I and III.

Periodic functions and trigonometric identities

• Trig functions repeat at regular intervals: sine and cosine have period $2\pi$, while tangent has period $\pi$. This periodicity is why general solutions to trig equations include terms like $+ 2\pi k$.
• The Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ is the foundation for many of the formulas in this section. The alternate cosine double-angle forms, for instance, come directly from substituting this identity.