Pre-Calculus: Sum and Difference Formulas

Mastering the sum and difference formulas is a pivotal step in pre-calculus, moving you from memorizing unit circle values to actively manipulating and combining trigonometric functions. These formulas are not just abstract exercises; they are essential tools for simplifying complex expressions, modeling periodic phenomena in engineering, and unlocking more advanced calculus concepts. By learning to break down angles like $75^{\circ }$ or $\frac{5\pi }{12}$ into sums of known angles, you gain the power to compute exact values and prove deeper mathematical truths.

The Foundation: What the Formulas State

The sum and difference formulas express the sine, cosine, and tangent of an angle that is the sum or difference of two other angles ($A\pm B$) in terms of the trigonometric functions of $A$ and $B$ individually. These identities are your algebraic toolkit for deconstructing complex angles.

For sine and cosine, the formulas are as follows:

For Sine: $$\sin (A+B)=\sin A\cos B+\cos A\sin B$$ $$\sin (A-B)=\sin A\cos B-\cos A\sin B$$

For Cosine: $$\cos (A+B)=\cos A\cos B-\sin A\sin B$$ $$\cos (A-B)=\cos A\cos B+\sin A\sin B$$

A helpful mnemonic for sine is "Sine mixes signs and cosines": $\sin (A\pm B)$ uses the same sign as the left-hand side between the terms. For cosine, remember "Cosine is cautious and keeps cosines together, but changes the sign": $\cos (A\pm B)$ uses the opposite sign between its terms.

The tangent formulas are derived from the sine and cosine identities: $$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$$ $$\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$$

These formulas hold true for any angles $A$ and $B$ where the functions are defined, making them universally applicable.

Deriving Double-Angle Formulas as a Special Case

One of the most powerful applications of the sum formulas is the immediate derivation of the double-angle formulas. If you set $B=A$ in the sum formulas, you create expressions for $\sin (2A)$, $\cos (2A)$, and $\tan (2A)$.

Starting with $\sin (A+A)$: $$\sin (A+A)=\sin A\cos A+\cos A\sin A$$ $$\sin (2A)=2\sin A\cos A$$

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

Using the Pythagorean Identity ${\sin }^{2}A+{\cos }^{2}A=1$, we can derive two other useful forms for $\cos (2A)$: $$\cos (2A)=2{\cos }^{2}A-1$$ $$\cos (2A)=1-2{\sin }^{2}A$$

You choose the form based on what's most convenient for the problem at hand, such as simplifying an expression or solving an equation. The double-angle formulas are not separate entities to memorize in isolation; they are a direct and logical consequence of the sum formulas.

Computing Exact Values for Non-Standard Angles

The primary computational use of these formulas is finding exact trigonometric values for angles not found on the standard unit circle. The strategy is to express your target angle as a sum or difference of two angles whose sine and cosine you know exactly from the unit circle (e.g., $30^{\circ }$, $45^{\circ }$, $60^{\circ }$, and their radian equivalents).

Example 1: Find $\sin (15^{\circ })$ We can express $15^{\circ }$ as $45^{\circ }-30^{\circ }$. $$\sin (45^{\circ }-30^{\circ })=\sin 45^{\circ }\cos 30^{\circ }-\cos 45^{\circ }\sin 30^{\circ }$$ $$=\left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)-\left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)$$ $$=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}=\frac{\sqrt{6}-\sqrt{2}}{4}$$

Example 2: Find $\cos (105^{\circ })$ Express $105^{\circ }$ as $60^{\circ }+45^{\circ }$. $$\cos (60^{\circ }+45^{\circ })=\cos 60^{\circ }\cos 45^{\circ }-\sin 60^{\circ }\sin 45^{\circ }$$ $$=\left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)-\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)$$ $$=\frac{\sqrt{2}}{4}-\frac{\sqrt{6}}{4}=\frac{\sqrt{2}-\sqrt{6}}{4}$$

This process is like trigonometric puzzle-solving. For engineering applications, such as analyzing alternating current circuits with phase shifts, these exact values provide precision that decimal approximations cannot.

Proving and Verifying Other Identities

Beyond computation, sum and difference formulas are indispensable for proving more complex trigonometric identities. The general approach is to start with the more complex side of the identity, apply the relevant formula to expand it, and then simplify using other fundamental identities (like Pythagorean or reciprocal identities) until it matches the simpler side.

Example: Prove the identity $\cos (x-y)\cos y-\sin (x-y)\sin y=\cos x$.

1. Start with the left-hand side (LHS). Apply the difference formulas:

$$\text{LHS}=(\cos x\cos y+\sin x\sin y)\cos y-(\sin x\cos y-\cos x\sin y)\sin y$$

1. Distribute carefully:

$$=\cos x{\cos }^{2}y+\sin x\sin y\cos y-\sin x\sin y\cos y+\cos x{\sin }^{2}y$$

1. Notice the middle terms cancel each other out ($+\sin x\sin y\cos y-\sin x\sin y\cos y=0$).

$$=\cos x{\cos }^{2}y+\cos x{\sin }^{2}y$$

1. Factor out $\cos x$:

$$=\cos x({\cos }^{2}y+{\sin }^{2}y)$$

1. Apply the Pythagorean Identity ${\cos }^{2}y+{\sin }^{2}y=1$:

$$=\cos x\cdot 1=\cos x$$

This matches the right-hand side (RHS), completing the proof. Systematic expansion and simplification, as shown here, is the key to successful verification.

Common Pitfalls

1. Sign Errors in the Formulas: This is the most frequent mistake. Remember that for $\sin (A-B)$, the sign between the terms is a minus. For $\cos (A-B)$, the sign between the terms is a plus. A quick mental check: $\cos (0)=\cos (x-x)$ should equal $\cos x\cos x+\sin x\sin x={\cos }^{2}x+{\sin }^{2}x=1$, which confirms the plus sign is correct.

1. Misapplying the Formulas to Coefficients: The formulas apply to the angles inside the function, not coefficients outside. For example, $\sin (2x+30^{\circ })$ uses $A=2x$ and $B=30^{\circ }$. However, $2\sin (x+30^{\circ })$ is simply $2$ times the expanded result of $\sin (x+30^{\circ })$. Do not bring the coefficient inside the angle argument.

1. Forgetting the Domain for Tangent Formulas: The tangent formulas $\frac{\tan A\pm \tan B}{1\mp \tan A\tan B}$ are only valid when $\tan A\tan B\ne \pm 1$ (to avoid division by zero) and when $\cos A\cos B\ne 0$ (since $\tan A$ and $\tan B$ must be defined). Always check that the angles involved do not violate these conditions.

1. Algebraic Mistakes During Simplification: When proving identities or computing exact values, errors in distribution, combining like terms, or applying Pythagorean identities are common. Work step-by-step, write legibly, and constantly look for opportunities to combine or cancel terms, as seen in the proof example above.

Summary

• The sum and difference formulas allow you to rewrite $\sin (A\pm B)$, $\cos (A\pm B)$, and $\tan (A\pm B)$ in terms of the functions of $A$ and $B$ alone, with careful attention to the signs connecting the terms.
• Setting $B=A$ in the sum formulas directly yields the essential double-angle formulas, such as $\sin (2A)=2\sin A\cos A$ and $\cos (2A)={\cos }^{2}A-{\sin }^{2}A$.
• The primary computational use is finding exact trigonometric values for non-standard angles by expressing them as sums or differences of known unit circle angles (e.g., $15^{\circ }=45^{\circ }-30^{\circ }$).
• These formulas serve as a foundational tool for proving more complex trigonometric identities through systematic expansion and simplification using basic algebraic rules and fundamental identities.
• Success hinges on memorizing the correct sign patterns, executing careful algebra, and understanding that these formulas manipulate the angle inputs to the function, not external coefficients.