Sum, Difference, and Double-Angle Formulas

The sum and difference formulas let you find exact trig values for angles that are not on the standard unit circle — angles like 75° (which is 45° + 30°) or 15° (which is 45° − 30°). By combining known values of special angles, you can compute exact results without a calculator. These identities are the foundation for the double-angle, half-angle, and product-to-sum formulas that appear throughout calculus, physics, and advanced standardized test questions. If you plan to study calculus or work in any technical field that involves wave analysis, these formulas are essential.

Sum and Difference Formulas

There are six formulas — two each for sine, cosine, and tangent.

Sine

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\sin(A - B) = \sin A \cos B - \cos A \sin B$

Cosine

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

$\cos(A - B) = \cos A \cos B + \sin A \sin B$

Tangent

$\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}$

$\tan(A - B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B}$

Sign Pattern Memory Aid

Pay close attention to the signs — they follow a consistent but slightly tricky pattern:

• Sine formulas match the operation sign. The sum formula $\sin(A + B)$ uses a plus between terms; the difference formula $\sin(A - B)$ uses a minus.
• Cosine formulas flip the sign. The sum formula $\cos(A + B)$ uses a minus between terms; the difference formula $\cos(A - B)$ uses a plus. This reversal is the most common source of errors.
• Tangent has the opposite sign in the denominator. The sum formula has a minus in the denominator ($1 - \tan A \tan B$); the difference formula has a plus ($1 + \tan A \tan B$).

Worked Example 1: Find the Exact Value of sin(75°)

Since 75° is not a special angle, decompose it: $75° = 45° + 30°$.

$\sin(75°) = \sin(45° + 30°) = \sin 45° \cos 30° + \cos 45° \sin 30°$

Substitute the known exact values:

$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

$= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$

Answer: $\sin(75°) = \dfrac{\sqrt{6} + \sqrt{2}}{4} \approx 0.9659$

Worked Example 2: Find the Exact Value of cos(15°)

Decompose: $15° = 45° - 30°$.

$\cos(15°) = \cos(45° - 30°) = \cos 45° \cos 30° + \sin 45° \sin 30°$

Note the plus sign — cosine difference flips to addition:

$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

$= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$

Answer: $\cos(15°) = \dfrac{\sqrt{6} + \sqrt{2}}{4} \approx 0.9659$

Notice that $\sin(75°) = \cos(15°)$. This confirms the co-function identity: $\sin\theta = \cos(90° - \theta)$.

Worked Example 3: Simplify sin(x + π)

Apply the sine sum formula with $A = x$ and $B = \pi$:

$\sin(x + \pi) = \sin x \cdot \cos\pi + \cos x \cdot \sin\pi$

Since $\cos\pi = -1$ and $\sin\pi = 0$:

$= \sin x \cdot (-1) + \cos x \cdot (0) = -\sin x$

Answer: $\sin(x + \pi) = -\sin x$

This result explains why sine is negative in Quadrant III — adding $\pi$ radians (180°) to any angle moves it to the opposite side of the unit circle, flipping the sign of sine.

Double-Angle Formulas

The double-angle formulas are a direct consequence of the sum formulas. Set $B = A$ in each sum formula, and you get the double-angle version.

Deriving sin(2θ)

Start with the sine sum formula:

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

Set $A = B = \theta$:

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

$\boxed{\sin(2\theta) = 2\sin\theta\cos\theta}$

Cosine Double-Angle (Three Forms)

Setting $A = B = \theta$ in the cosine sum formula:

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

This primary form can be rewritten two more ways using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$:

Form 2: Substitute $\sin^2\theta = 1 - \cos^2\theta$:

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

Form 3: Substitute $\cos^2\theta = 1 - \sin^2\theta$:

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

All three are the same identity in different clothing:

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

Choose whichever form matches the information you have. If you know only cosine, use $2\cos^2\theta - 1$. If you know only sine, use $1 - 2\sin^2\theta$.

Tangent Double-Angle

Setting $A = B = \theta$ in the tangent sum formula:

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

Worked Example 4: Find sin(2θ) Given sinθ = 3/5 in Quadrant I

First, find $\cos\theta$ using the Pythagorean identity:

$\cos^2\theta = 1 - \sin^2\theta = 1 - \frac{9}{25} = \frac{16}{25}$

Since $\theta$ is in Quadrant I, cosine is positive: $\cos\theta = \frac{4}{5}$.

Now apply the double-angle formula:

$\sin(2\theta) = 2\sin\theta\cos\theta = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}$

Answer: $\sin(2\theta) = \dfrac{24}{25}$

Worked Example 5: Verify the cos(2θ) Identity

Show that $\cos(2\theta) = 1 - 2\sin^2\theta$ follows from $\cos(2\theta) = \cos^2\theta - \sin^2\theta$.

Start with the primary form:

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

Substitute $\cos^2\theta = 1 - \sin^2\theta$ (Pythagorean identity rearranged):

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

This confirms that the third form is equivalent. You can verify the second form ($2\cos^2\theta - 1$) by substituting $\sin^2\theta = 1 - \cos^2\theta$ instead.

Real-World Connection: AC Circuits and Signal Processing

In alternating current (AC) electrical circuits, both voltage and current are sinusoidal. The instantaneous power involves products like $\sin(\omega t) \cdot \sin(\omega t + \varphi)$, where $\varphi$ is the phase angle. Analyzing this product requires the sum formula to expand $\sin(\omega t + \varphi)$, which lets engineers decompose the power into real (useful) power and reactive power.

In signal processing, these formulas are the mathematical basis for frequency mixing and modulation — the techniques used to transmit radio, TV, and cell phone signals. When two signals at different frequencies are multiplied together, the sum and difference formulas show that the result contains signals at the sum and difference of the original frequencies.

Common Mistakes

1. Writing $\sin(A + B) = \sin A + \sin B$ — This is wrong. You cannot distribute the sine function over addition. The sine of a sum requires the full four-term expansion.
2. Getting the sign wrong in cosine formulas — Remember that cosine flips the sign: $\cos(A + B)$ uses subtraction, and $\cos(A - B)$ uses addition. This is the opposite of what you might expect.
3. Forgetting to use exact values from special angles — When computing $\sin(75°)$ or $\cos(15°)$, use the exact values $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$, and $\frac{1}{2}$ from the special angles table. Decimal approximations defeat the purpose.
4. Confusing when to use sum formulas vs double-angle formulas — The double-angle formulas are for $\sin(2\theta)$, $\cos(2\theta)$, etc. — always twice the same angle. The sum and difference formulas are for combining two different angles.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The sum and difference formulas expand $\sin(A \pm B)$, $\cos(A \pm B)$, and $\tan(A \pm B)$ using only the trig values of $A$ and $B$ individually
• Sine matches the operation sign; cosine flips it; tangent puts the opposite sign in the denominator
• These formulas let you compute exact values for non-special angles like 15°, 75°, 105°, and 165°
• The double-angle formulas are the special case where $B = A$, giving $\sin(2\theta) = 2\sin\theta\cos\theta$ and three equivalent forms for $\cos(2\theta)$
• Choose the form of $\cos(2\theta)$ that matches what you know: use $2\cos^2\theta - 1$ if you know cosine, or $1 - 2\sin^2\theta$ if you know sine
• These identities appear throughout calculus (integration techniques), physics (wave superposition), and electrical engineering (AC power analysis)

Return to Trigonometry for more topics in this section.