Proving Angle Addition/Subtraction Formulas - Precalculus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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