Proving Angle Addition/Subtraction Formulas - Pre-Calculus
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State the formula for $\sin(\alpha+\beta)$ in terms of $\sin$ and $\cos$.
State the formula for $\sin(\alpha+\beta)$ in terms of $\sin$ and $\cos$.
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$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Expands using the product of sine and cosine terms with matching signs.
$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Expands using the product of sine and cosine terms with matching signs.
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Identify the restriction needed for $\tan(\alpha+\beta)$ to be defined in its fraction form.
Identify the restriction needed for $\tan(\alpha+\beta)$ to be defined in its fraction form.
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$1-\tan\alpha\tan\beta\ne^0$. The denominator cannot equal zero for the fraction to be defined.
$1-\tan\alpha\tan\beta\ne^0$. The denominator cannot equal zero for the fraction to be defined.
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Find $\cos(2\theta)$ in terms of $\cos\theta$ and $\sin\theta$ using $\cos(\alpha+\beta)$.
Find $\cos(2\theta)$ in terms of $\cos\theta$ and $\sin\theta$ using $\cos(\alpha+\beta)$.
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$\cos(2\theta)=\cos^2\theta-\sin^2\theta$. Set $\alpha=\beta=\theta$ in the cosine addition formula.
$\cos(2\theta)=\cos^2\theta-\sin^2\theta$. Set $\alpha=\beta=\theta$ in the cosine addition formula.
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Find $\sin(2\theta)$ in terms of $\sin\theta$ and $\cos\theta$ using $\sin(\alpha+\beta)$.
Find $\sin(2\theta)$ in terms of $\sin\theta$ and $\cos\theta$ using $\sin(\alpha+\beta)$.
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$\sin(2\theta)=2\sin\theta\cos\theta$. Set $\alpha=\beta=\theta$ in the sine addition formula.
$\sin(2\theta)=2\sin\theta\cos\theta$. Set $\alpha=\beta=\theta$ in the sine addition formula.
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Use angle addition to find $\tan\left(\frac{5\pi}{12}\right)$ exactly.
Use angle addition to find $\tan\left(\frac{5\pi}{12}\right)$ exactly.
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$2+\sqrt{3}$. Write $\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}$ and apply tangent addition formula.
$2+\sqrt{3}$. Write $\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}$ and apply tangent addition formula.
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Use angle addition to find $\tan\left(\frac{\pi}{12}\right)$ exactly.
Use angle addition to find $\tan\left(\frac{\pi}{12}\right)$ exactly.
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$2-\sqrt{3}$. Write $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$ and apply tangent subtraction formula.
$2-\sqrt{3}$. Write $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$ and apply tangent subtraction formula.
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Use angle subtraction to find $\cos\left(\frac{\pi}{12}\right)$ exactly.
Use angle subtraction to find $\cos\left(\frac{\pi}{12}\right)$ exactly.
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$\frac{\sqrt{6}+\sqrt{2}}{4}$. Write $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$ and apply cosine subtraction formula.
$\frac{\sqrt{6}+\sqrt{2}}{4}$. Write $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$ and apply cosine subtraction formula.
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Use angle subtraction to find $\sin\left(\frac{\pi}{12}\right)$ exactly.
Use angle subtraction to find $\sin\left(\frac{\pi}{12}\right)$ exactly.
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$\frac{\sqrt{6}-\sqrt{2}}{4}$. Write $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$ and apply sine subtraction formula.
$\frac{\sqrt{6}-\sqrt{2}}{4}$. Write $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$ and apply sine subtraction formula.
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Use angle addition to find $\cos\left(\frac{5\pi}{12}\right)$ exactly.
Use angle addition to find $\cos\left(\frac{5\pi}{12}\right)$ exactly.
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$\frac{\sqrt{6}-\sqrt{2}}{4}$. Write $\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}$ and apply cosine addition formula.
$\frac{\sqrt{6}-\sqrt{2}}{4}$. Write $\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}$ and apply cosine addition formula.
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Use angle addition to find $\sin\left(\frac{5\pi}{12}\right)$ exactly.
Use angle addition to find $\sin\left(\frac{5\pi}{12}\right)$ exactly.
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$\frac{\sqrt{6}+\sqrt{2}}{4}$. Write $\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}$ and apply sine addition formula.
$\frac{\sqrt{6}+\sqrt{2}}{4}$. Write $\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}$ and apply sine addition formula.
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Find $\tan(\alpha-\beta)$ if $\tan\alpha=2$ and $\tan\beta=\frac{1}{3}$.
Find $\tan(\alpha-\beta)$ if $\tan\alpha=2$ and $\tan\beta=\frac{1}{3}$.
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$1$. Use tangent subtraction: $\frac{2-\frac{1}{3}}{1+2\cdot\frac{1}{3}}=\frac{\frac{5}{3}}{\frac{5}{3}}=1$.
$1$. Use tangent subtraction: $\frac{2-\frac{1}{3}}{1+2\cdot\frac{1}{3}}=\frac{\frac{5}{3}}{\frac{5}{3}}=1$.
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Find $\tan(\alpha+\beta)$ if $\tan\alpha=\frac{1}{2}$ and $\tan\beta=\frac{1}{3}$.
Find $\tan(\alpha+\beta)$ if $\tan\alpha=\frac{1}{2}$ and $\tan\beta=\frac{1}{3}$.
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$1$. Use tangent addition: $\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2}\cdot\frac{1}{3}}=\frac{\frac{5}{6}}{\frac{5}{6}}=1$.
$1$. Use tangent addition: $\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2}\cdot\frac{1}{3}}=\frac{\frac{5}{6}}{\frac{5}{6}}=1$.
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What is $\cos(\alpha+\beta)$ if $\cos\alpha=\frac{4}{5}$, $\sin\alpha=\frac{3}{5}$, $\cos\beta=\frac{12}{13}$, $\sin\beta=\frac{5}{13}$?
What is $\cos(\alpha+\beta)$ if $\cos\alpha=\frac{4}{5}$, $\sin\alpha=\frac{3}{5}$, $\cos\beta=\frac{12}{13}$, $\sin\beta=\frac{5}{13}$?
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$\frac{33}{65}$. Apply $\cos(\alpha+\beta)$ formula: $\frac{4}{5}\cdot\frac{12}{13}-\frac{3}{5}\cdot\frac{5}{13}=\frac{33}{65}$.
$\frac{33}{65}$. Apply $\cos(\alpha+\beta)$ formula: $\frac{4}{5}\cdot\frac{12}{13}-\frac{3}{5}\cdot\frac{5}{13}=\frac{33}{65}$.
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What is $\sin(\alpha+\beta)$ if $\sin\alpha=\frac{3}{5}$, $\cos\alpha=\frac{4}{5}$, $\sin\beta=\frac{5}{13}$, $\cos\beta=\frac{12}{13}$?
What is $\sin(\alpha+\beta)$ if $\sin\alpha=\frac{3}{5}$, $\cos\alpha=\frac{4}{5}$, $\sin\beta=\frac{5}{13}$, $\cos\beta=\frac{12}{13}$?
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$\frac{56}{65}$. Apply $\sin(\alpha+\beta)$ formula: $\frac{3}{5}\cdot\frac{12}{13}+\frac{4}{5}\cdot\frac{5}{13}=\frac{56}{65}$.
$\frac{56}{65}$. Apply $\sin(\alpha+\beta)$ formula: $\frac{3}{5}\cdot\frac{12}{13}+\frac{4}{5}\cdot\frac{5}{13}=\frac{56}{65}$.
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Identify the identity used to convert a tangent sum into sine and cosine: $\tan\theta=?$
Identify the identity used to convert a tangent sum into sine and cosine: $\tan\theta=?$
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$\tan\theta=\frac{\sin\theta}{\cos\theta}$. Tangent equals sine divided by cosine.
$\tan\theta=\frac{\sin\theta}{\cos\theta}$. Tangent equals sine divided by cosine.
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State the formula for $\tan(\alpha-\beta)$ in terms of $\tan\alpha$ and $\tan\beta$.
State the formula for $\tan(\alpha-\beta)$ in terms of $\tan\alpha$ and $\tan\beta$.
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$\tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$. Difference of tangents over one plus their product.
$\tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$. Difference of tangents over one plus their product.
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State the formula for $\tan(\alpha+\beta)$ in terms of $\tan\alpha$ and $\tan\beta$.
State the formula for $\tan(\alpha+\beta)$ in terms of $\tan\alpha$ and $\tan\beta$.
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$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$. Sum of tangents over one minus their product.
$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$. Sum of tangents over one minus their product.
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State the formula for $\cos(\alpha-\beta)$ in terms of $\sin$ and $\cos$.
State the formula for $\cos(\alpha-\beta)$ in terms of $\sin$ and $\cos$.
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$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$. Same as addition but with a plus sign between the products.
$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$. Same as addition but with a plus sign between the products.
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State the formula for $\cos(\alpha+\beta)$ in terms of $\sin$ and $\cos$.
State the formula for $\cos(\alpha+\beta)$ in terms of $\sin$ and $\cos$.
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$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. Products of like functions minus products of unlike functions.
$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. Products of like functions minus products of unlike functions.
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State the formula for $\sin(\alpha-\beta)$ in terms of $\sin$ and $\cos$.
State the formula for $\sin(\alpha-\beta)$ in terms of $\sin$ and $\cos$.
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$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$. Same as addition but with a minus sign between the products.
$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$. Same as addition but with a minus sign between the products.
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Identify the identity that expresses $\tan\theta$ using $\sin\theta$ and $\cos\theta$.
Identify the identity that expresses $\tan\theta$ using $\sin\theta$ and $\cos\theta$.
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$\tan\theta=\frac{\sin\theta}{\cos\theta}$. Tangent is the ratio of sine to cosine.
$\tan\theta=\frac{\sin\theta}{\cos\theta}$. Tangent is the ratio of sine to cosine.
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Identify the identity used to rewrite $\sin(-\theta)$ in terms of $\sin\theta$.
Identify the identity used to rewrite $\sin(-\theta)$ in terms of $\sin\theta$.
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$\sin(-\theta)=-\sin\theta$. Sine is an odd function, so negating the angle negates the value.
$\sin(-\theta)=-\sin\theta$. Sine is an odd function, so negating the angle negates the value.
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Identify the identity used to rewrite $\cos(-\theta)$ in terms of $\cos\theta$.
Identify the identity used to rewrite $\cos(-\theta)$ in terms of $\cos\theta$.
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$\cos(-\theta)=\cos\theta$. Cosine is an even function, so negating the angle doesn't change the value.
$\cos(-\theta)=\cos\theta$. Cosine is an even function, so negating the angle doesn't change the value.
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Identify the condition on $\alpha$ and $\beta$ required for $\tan(\alpha+\beta)$ to be defined.
Identify the condition on $\alpha$ and $\beta$ required for $\tan(\alpha+\beta)$ to be defined.
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$1-\tan\alpha\tan\beta\neq 0$. The denominator must be nonzero to avoid division by zero.
$1-\tan\alpha\tan\beta\neq 0$. The denominator must be nonzero to avoid division by zero.
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Identify the identity used to rewrite $\tan(-\theta)$ in terms of $\tan\theta$.
Identify the identity used to rewrite $\tan(-\theta)$ in terms of $\tan\theta$.
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$\tan(-\theta)=-\tan\theta$. Tangent is an odd function, so negating the angle negates the value.
$\tan(-\theta)=-\tan\theta$. Tangent is an odd function, so negating the angle negates the value.
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Evaluate $\sin(75^\circ)$ exactly using an angle addition identity.
Evaluate $\sin(75^\circ)$ exactly using an angle addition identity.
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$\frac{\sqrt{6}+\sqrt{2}}{4}$. Use $\sin(75°)=\sin(45°+30°)$ with addition formula.
$\frac{\sqrt{6}+\sqrt{2}}{4}$. Use $\sin(75°)=\sin(45°+30°)$ with addition formula.
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Evaluate $\cos(75^\circ)$ exactly using an angle addition identity.
Evaluate $\cos(75^\circ)$ exactly using an angle addition identity.
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$\frac{\sqrt{6}-\sqrt{2}}{4}$. Apply $\cos(75°)=\cos(45°+30°)$ using addition identity.
$\frac{\sqrt{6}-\sqrt{2}}{4}$. Apply $\cos(75°)=\cos(45°+30°)$ using addition identity.
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State the rotation matrix $R(\theta)$ used in a common proof of angle addition formulas.
State the rotation matrix $R(\theta)$ used in a common proof of angle addition formulas.
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$R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\sin\theta&\cos\theta\end{pmatrix}$. Standard 2D rotation matrix for counterclockwise rotation.
$R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\sin\theta&\cos\theta\end{pmatrix}$. Standard 2D rotation matrix for counterclockwise rotation.
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What denominator condition must hold for $\tan(\alpha+\beta)$ to be defined from its formula?
What denominator condition must hold for $\tan(\alpha+\beta)$ to be defined from its formula?
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$1-\tan\alpha\tan\beta\neq 0$. Ensures the denominator is non-zero for the formula to exist.
$1-\tan\alpha\tan\beta\neq 0$. Ensures the denominator is non-zero for the formula to exist.
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Identify the identity used to derive $\tan(\alpha\pm\beta)$ from $\sin$ and $\cos$ formulas.
Identify the identity used to derive $\tan(\alpha\pm\beta)$ from $\sin$ and $\cos$ formulas.
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$\tan\theta=\frac{\sin\theta}{\cos\theta}$. Fundamental identity relating tangent to sine and cosine.
$\tan\theta=\frac{\sin\theta}{\cos\theta}$. Fundamental identity relating tangent to sine and cosine.
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