Precalculus Flashcards: Proving Angle Addition Subtraction Formulas

What this deck covers

This deck focuses on Proving Angle Addition Subtraction Formulas, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find $\tan(\alpha-\beta)$ if $\tan\alpha=2$ and $\tan\beta=\frac{1}{3}$.

Answer: $1$. Use tangent subtraction: $\frac{2-\frac{1}{3}}{1+2\cdot\frac{1}{3}}=\frac{\frac{5}{3}}{\frac{5}{3}}=1$.

Flashcard 2: Find $\tan(\alpha+\beta)$ if $\tan\alpha=\frac{1}{2}$ and $\tan\beta=\frac{1}{3}$.

Answer: $1$. Use tangent addition: $\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2}\cdot\frac{1}{3}}=\frac{\frac{5}{6}}{\frac{5}{6}}=1$.

Flashcard 3: What is $\cos(\alpha+\beta)$ if $\cos\alpha=\frac{4}{5}$, $\sin\alpha=\frac{3}{5}$, $\cos\beta=\frac{12}{13}$, $\sin\beta=\frac{5}{13}$?

Answer: $\frac{33}{65}$. Apply $\cos(\alpha+\beta)$ formula: $\frac{4}{5}\cdot\frac{12}{13}-\frac{3}{5}\cdot\frac{5}{13}=\frac{33}{65}$.

Flashcard 4: State the rotation matrix $R(\theta)$ used in a common proof of angle addition formulas.

Answer: $R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\sin\theta&\cos\theta\end{pmatrix}$. Standard 2D rotation matrix for counterclockwise rotation.

Flashcard 5: What denominator condition must hold for $\tan(\alpha-\beta)$ to be defined from its formula?

Answer: $1+\tan\alpha\tan\beta\neq 0$. Prevents division by zero in the tangent difference formula.

Flashcard 6: Use angle subtraction to find $\cos\left(\frac{\pi}{12}\right)$ exactly.

Answer: $\frac{\sqrt{6}+\sqrt{2}}{4}$. Write $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$ and apply cosine subtraction formula.

Flashcard 7: State the formula for $\cos(\alpha+\beta)$ in terms of $\sin$ and $\cos$.

Answer: $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. Products of like functions minus products of unlike functions.

Flashcard 8: Identify the identity used to rewrite $\tan(-\theta)$ in terms of $\tan\theta$.

Answer: $\tan(-\theta)=-\tan\theta$. Tangent is an odd function, so negating the angle negates the value.

Flashcard 9: Find $\tan\left(\frac{5\pi}{12}\right)$ using an addition formula.

Answer: $2+\sqrt{3}$. Apply $\tan(\frac{5\pi}{12})=\tan(\frac{\pi}{4}+\frac{\pi}{6})$ using addition.

Flashcard 10: State the formula for $\sin(\alpha+\beta)$ in terms of $\sin$ and $\cos$.

Answer: $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Expands using the product of sine and cosine terms with matching signs.

Flashcard 11: State the formula for $\tan(\alpha+\beta)$ in terms of $\tan\alpha$ and $\tan\beta$.

Answer: $\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$. Sum of tangents over one minus their product.

Flashcard 12: Evaluate $\cos(15^\circ)$ exactly using an angle subtraction identity.

Answer: $\frac{\sqrt{6}+\sqrt{2}}{4}$. Apply $\cos(15°)=\cos(45°-30°)$ using subtraction identity.

Flashcard 13: Find $\sin(2\theta)$ in terms of $\sin\theta$ and $\cos\theta$ using $\sin(\alpha+\beta)$.

Answer: $\sin(2\theta)=2\sin\theta\cos\theta$. Set $\alpha=\beta=\theta$ in the sine addition formula.

Flashcard 14: Evaluate $\cos(75^\circ)$ exactly using an angle addition identity.

Answer: $\frac{\sqrt{6}-\sqrt{2}}{4}$. Apply $\cos(75°)=\cos(45°+30°)$ using addition identity.

Flashcard 15: Evaluate $\sin(15^\circ)$ exactly using an angle subtraction identity.

Answer: $\frac{\sqrt{6}-\sqrt{2}}{4}$. Use $\sin(15°)=\sin(45°-30°)$ with subtraction formula.

Flashcard 16: What is $\sin(\alpha+\beta)$ if $\sin\alpha=\frac{3}{5}$, $\cos\alpha=\frac{4}{5}$, $\sin\beta=\frac{5}{13}$, $\cos\beta=\frac{12}{13}$?

Answer: $\frac{56}{65}$. Apply $\sin(\alpha+\beta)$ formula: $\frac{3}{5}\cdot\frac{12}{13}+\frac{4}{5}\cdot\frac{5}{13}=\frac{56}{65}$.

Flashcard 17: Use angle addition to find $\cos\left(\frac{5\pi}{12}\right)$ exactly.

Answer: $\frac{\sqrt{6}-\sqrt{2}}{4}$. Write $\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}$ and apply cosine addition formula.

Flashcard 18: Use angle subtraction to find $\sin\left(\frac{\pi}{12}\right)$ exactly.

Answer: $\frac{\sqrt{6}-\sqrt{2}}{4}$. Write $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$ and apply sine subtraction formula.

Flashcard 19: Identify the identity used to rewrite $\cos(-\theta)$ in terms of $\cos\theta$.

Answer: $\cos(-\theta)=\cos\theta$. Cosine is an even function, so negating the angle doesn't change the value.

Flashcard 20: Identify the identity that expresses $\tan\theta$ using $\sin\theta$ and $\cos\theta$.

Answer: $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Tangent is the ratio of sine to cosine.

Flashcard 21: Identify the identity used to rewrite $\sin(-\theta)$ in terms of $\sin\theta$.

Answer: $\sin(-\theta)=-\sin\theta$. Sine is an odd function, so negating the angle negates the value.

Flashcard 22: Use angle addition to find $\tan\left(\frac{5\pi}{12}\right)$ exactly.

Answer: $2+\sqrt{3}$. Write $\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}$ and apply tangent addition formula.

Flashcard 23: State the formula for $\cos(\alpha-\beta)$ in terms of $\sin$ and $\cos$.

Answer: $\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$. Same as addition but with a plus sign between the products.

Flashcard 24: Find $\sin\left(\frac{5\pi}{12}\right)$ using an addition formula.

Answer: $\frac{\sqrt{6}+\sqrt{2}}{4}$. Use $\sin(\frac{5\pi}{12})=\sin(\frac{\pi}{4}+\frac{\pi}{6})$ with addition formula.

Flashcard 25: Identify the restriction needed for $\tan(\alpha+\beta)$ to be defined in its fraction form.

Answer: $1-\tan\alpha\tan\beta\ne^0$. The denominator cannot equal zero for the fraction to be defined.

Flashcard 26: Evaluate $\sin(75^\circ)$ exactly using an angle addition identity.

Answer: $\frac{\sqrt{6}+\sqrt{2}}{4}$. Use $\sin(75°)=\sin(45°+30°)$ with addition formula.

Flashcard 27: Find and correct the sign error: $\cos(\alpha+\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$.

Answer: $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. The sine product term must be subtracted, not added.

Flashcard 28: Identify the identity used to derive $\tan(\alpha\pm\beta)$ from $\sin$ and $\cos$ formulas.

Answer: $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Fundamental identity relating tangent to sine and cosine.

Flashcard 29: State the formula for $\tan(\alpha-\beta)$ in terms of $\tan\alpha$ and $\tan\beta$.

Answer: $\tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$. Difference of tangents over one plus their product.

Flashcard 30: Identify the key matrix equation used to prove addition formulas via rotations.

Answer: $R(\alpha)R(\beta)=R(\alpha+\beta)$. Composition of rotations equals rotation by sum of angles.

Flashcard 31: State the formula for $\sin(\alpha-\beta)$ in terms of $\sin$ and $\cos$.

Answer: $\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$. Same as addition but with a minus sign between the products.

Flashcard 32: Find $\cos\left(\frac{5\pi}{12}\right)$ using an addition formula.

Answer: $\frac{\sqrt{6}-\sqrt{2}}{4}$. Apply $\cos(\frac{5\pi}{12})=\cos(\frac{\pi}{4}+\frac{\pi}{6})$ using addition.

Flashcard 33: Identify the condition on $\alpha$ and $\beta$ required for $\tan(\alpha+\beta)$ to be defined.

Answer: $1-\tan\alpha\tan\beta\neq 0$. The denominator must be nonzero to avoid division by zero.

Flashcard 34: Use angle addition to find $\sin\left(\frac{5\pi}{12}\right)$ exactly.

Answer: $\frac{\sqrt{6}+\sqrt{2}}{4}$. Write $\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}$ and apply sine addition formula.

Flashcard 35: Use angle addition to find $\tan\left(\frac{\pi}{12}\right)$ exactly.

Answer: $2-\sqrt{3}$. Write $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$ and apply tangent subtraction formula.

Flashcard 36: Find $\cos(2\theta)$ in terms of $\cos\theta$ and $\sin\theta$ using $\cos(\alpha+\beta)$.

Answer: $\cos(2\theta)=\cos^2\theta-\sin^2\theta$. Set $\alpha=\beta=\theta$ in the cosine addition formula.

Flashcard 37: Identify the identity used to convert a tangent sum into sine and cosine: $\tan\theta=?$

Answer: $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Tangent equals sine divided by cosine.

Flashcard 38: What denominator condition must hold for $\tan(\alpha+\beta)$ to be defined from its formula?

Answer: $1-\tan\alpha\tan\beta\neq 0$. Ensures the denominator is non-zero for the formula to exist.

Flashcard 39: Find $\tan\left(\frac{\pi}{12}\right)$ using a subtraction formula.

Answer: $2-\sqrt{3}$. Use $\tan(\frac{\pi}{12})=\tan(\frac{\pi}{3}-\frac{\pi}{4})$ with subtraction.