Trigonometric Addition Formulae

Sine Addition Formulae

• These identities relate the sine of a sum or difference of two angles to the sines and cosines of the individual angles. The sign on the left-hand side (LHS) of the formula directly matches the sign on the right-hand side (RHS).

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

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

Cosine Addition Formulae

• These identities express the cosine of a sum or difference of two angles. A key characteristic is that the sign on the LHS is always opposite to the sign on the RHS, which is a common point of error if not carefully noted.

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

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

Tangent Addition Formulae

• These identities provide expressions for the tangent of a sum or difference of two angles. For tangent, the sign in the numerator matches the sign on the LHS, while the sign in the denominator is opposite.

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

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

• Double Angle Formulae are a special case of the addition formulae, derived by setting the two angles $A$ and $B$ equal to each other (i.e., $B=A$). These formulae are frequently used in calculus and advanced trigonometry but are generally not provided on exam formula sheets, requiring students to either memorize them or derive them quickly.

• For sine, setting $B=A$ in $\sin(A+B)$ yields the double angle formula for sine. This formula is unique and straightforward, expressing $\sin(2A)$ in terms of $\sin A$ and $\cos A$.

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

• For cosine, setting $B=A$ in $\cos(A+B)$ initially gives $\cos(2A) = \cos^2 A - \sin^2 A$. This form can then be further manipulated using the Pythagorean identity ($\sin^2 A + \cos^2 A = 1$) to produce two alternative forms, which are often more useful depending on the context of the problem.

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

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

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

• For tangent, setting $B=A$ in $\tan(A+B)$ results in the double angle formula for tangent. This formula is particularly useful when dealing with expressions involving $\tan(2A)$.

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

• Calculating Exact Values: One primary application is to find the exact trigonometric values for angles that are sums or differences of common angles (e.g., $15^\circ = 45^\circ - 30^\circ$, $75^\circ = 45^\circ + 30^\circ$). By expressing the target angle as a combination of angles with known exact trigonometric values, the formulae allow for precise calculation without a calculator.

• Simplifying Expressions and Proving Identities: The addition and double angle formulae are indispensable tools for simplifying complex trigonometric expressions into more manageable forms or for proving other trigonometric identities. This often involves recognizing patterns and strategically applying the appropriate formula to transform one side of an identity into the other.

• Solving Trigonometric Equations: In many cases, trigonometric equations can be simplified or transformed into a solvable form by applying these formulae. For instance, an equation involving $\sin(2x)$ might be converted to $2 \sin x \cos x$ to allow for factorization or substitution.

• Addition Formulae vs. Double Angle Formulae: Addition formulae are general for any two angles $A$ and $B$, while double angle formulae are a specific case where $A=B$. The double angle formulae are derived directly from the addition formulae, highlighting their interconnectedness.

• Relationship with Pythagorean Identity: The Pythagorean identity ($\sin^2 \theta + \cos^2 \theta = 1$) is crucial for manipulating the cosine double angle formula. It allows for the transformation of $\cos(2A) = \cos^2 A - \sin^2 A$ into forms involving only $\sin A$ or only $\cos A$, which is often necessary for solving specific problems.

• Sign Conventions: A critical distinction lies in the sign conventions for sine, cosine, and tangent addition formulae. Sine formulae maintain the sign, cosine formulae reverse it, and tangent formulae match the numerator but reverse the denominator. Understanding these patterns is key to avoiding errors.

• Incorrect Sign Usage: The most frequent error is misapplying the sign conventions, especially for cosine and tangent formulae. Students often assume the sign on the RHS always matches the LHS, leading to incorrect results. Always double-check the specific sign rules for each function.

• Confusing Double Angle Formulae: While derivable, students often confuse the three forms of $\cos(2A)$ or forget the $\sin(2A)$ and $\tan(2A)$ formulae. It's important to practice deriving them or commit them to memory to ensure quick and accurate application.

• Assuming Linearity: A common misconception is to assume that $\sin(A+B) = \sin A + \sin B$ or $\cos(A+B) = \cos A + \cos B$. Trigonometric functions are not linear in this way, and these incorrect assumptions lead to fundamental errors. The addition formulae explicitly show the non-linear relationship.

• Ignoring Domain Restrictions for Tangent: When using tangent formulae, it's important to remember that $\tan A$, $\tan B$, and $\tan(A \pm B)$ are undefined for angles that are odd multiples of $\pi/2$ ($90^\circ$). Also, the denominator of the tangent addition formulae cannot be zero, which implies $1 - \tan A \tan B \neq 0$ for $\tan(A+B)$ and $1 + \tan A \tan B \neq 0$ for $\tan(A-B)$.

• Memorize Sign Conventions: Even though the formulae are often provided, internalizing the sign patterns for sine, cosine, and tangent addition formulae will save time and prevent errors under exam pressure. Practice writing them out until they are second nature.

• Derive Double Angle Formulae: If you struggle to remember the double angle formulae, practice deriving them from the addition formulae. This skill ensures you can always access them, even if memory fails, and reinforces understanding of their origin.

• Check for Exact Values: When asked for exact values, immediately consider if the angle can be expressed as a sum or difference of $30^\circ, 45^\circ, 60^\circ$ (or $\pi/6, \pi/4, \pi/3$). This is a strong indicator that addition formulae are required.

• Work Systematically: For proofs or complex simplifications, work step-by-step, applying one identity at a time. Clearly show each transformation and keep the target expression in mind. If stuck, try working from the other side of the identity or trying a different form of a double angle formula.