Proving Angle Addition/Subtraction Formulas - Geometry
Card 1 of 30
State the formula for $\cos(\alpha+\beta)$.
State the formula for $\cos(\alpha+\beta)$.
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$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. Standard angle addition formula for cosine function.
$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. Standard angle addition formula for cosine function.
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Identify the correct identity for $\cos(\alpha+\beta)$ among sign choices before $\sin\alpha\sin\beta$.
Identify the correct identity for $\cos(\alpha+\beta)$ among sign choices before $\sin\alpha\sin\beta$.
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$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. The correct sign before $\sin\alpha\sin\beta$ in cosine addition is negative.
$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. The correct sign before $\sin\alpha\sin\beta$ in cosine addition is negative.
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State the formula for $\sin(\alpha+\beta)$.
State the formula for $\sin(\alpha+\beta)$.
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$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Standard angle addition formula for sine function.
$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Standard angle addition formula for sine function.
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State the formula for $\sin(\alpha-\beta)$.
State the formula for $\sin(\alpha-\beta)$.
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$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$. Standard angle subtraction formula for sine function.
$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$. Standard angle subtraction formula for sine function.
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State the formula for $\cos(\alpha-\beta)$.
State the formula for $\cos(\alpha-\beta)$.
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$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$. Standard angle subtraction formula for cosine function.
$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$. Standard angle subtraction formula for cosine function.
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State the formula for $\tan(\alpha+\beta)$.
State the formula for $\tan(\alpha+\beta)$.
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$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$. Standard angle addition formula for tangent function.
$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$. Standard angle addition formula for tangent function.
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State the formula for $\tan(\alpha-\beta)$.
State the formula for $\tan(\alpha-\beta)$.
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$\tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$. Standard angle subtraction formula for tangent function.
$\tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$. Standard angle subtraction formula for tangent function.
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Identify the Pythagorean identity relating $\sin\theta$ and $\cos\theta$.
Identify the Pythagorean identity relating $\sin\theta$ and $\cos\theta$.
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$\sin^2\theta+\cos^2\theta=1$. Fundamental Pythagorean identity for unit circle.
$\sin^2\theta+\cos^2\theta=1$. Fundamental Pythagorean identity for unit circle.
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What identity defines tangent in terms of sine and cosine?
What identity defines tangent in terms of sine and cosine?
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$\tan\theta=\frac{\sin\theta}{\cos\theta}$. Fundamental quotient identity relating tangent to sine and cosine.
$\tan\theta=\frac{\sin\theta}{\cos\theta}$. Fundamental quotient identity relating tangent to sine and cosine.
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State the cofunction identity for tangent: what is $\tan\left(\frac{\pi}{2}-\theta\right)$?
State the cofunction identity for tangent: what is $\tan\left(\frac{\pi}{2}-\theta\right)$?
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$\tan\left(\frac{\pi}{2}-\theta\right)=\frac{1}{\tan\theta}$. Tangent cofunction identity equivalent to $\cot\theta$.
$\tan\left(\frac{\pi}{2}-\theta\right)=\frac{1}{\tan\theta}$. Tangent cofunction identity equivalent to $\cot\theta$.
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State the double-angle identity for sine: what is $\sin(2\theta)$?
State the double-angle identity for sine: what is $\sin(2\theta)$?
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$\sin(2\theta)=2\sin\theta\cos\theta$. Double-angle formula derived from sine addition formula.
$\sin(2\theta)=2\sin\theta\cos\theta$. Double-angle formula derived from sine addition formula.
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State one standard form of the double-angle identity for cosine: what is $\cos(2\theta)$?
State one standard form of the double-angle identity for cosine: what is $\cos(2\theta)$?
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$\cos(2\theta)=\cos^2\theta-\sin^2\theta$. Double-angle formula derived from cosine addition formula.
$\cos(2\theta)=\cos^2\theta-\sin^2\theta$. Double-angle formula derived from cosine addition formula.
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State the double-angle identity for tangent: what is $\tan(2\theta)$?
State the double-angle identity for tangent: what is $\tan(2\theta)$?
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$\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}$. Double-angle formula derived from tangent addition formula.
$\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}$. Double-angle formula derived from tangent addition formula.
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Identify the expression equivalent to $\sin(\beta-\alpha)$ in terms of $\sin(\alpha-\beta)$.
Identify the expression equivalent to $\sin(\beta-\alpha)$ in terms of $\sin(\alpha-\beta)$.
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$\sin(\beta-\alpha)=-\sin(\alpha-\beta)$. Sine is an odd function, so $\sin(\beta-\alpha) = -\sin(\alpha-\beta)$.
$\sin(\beta-\alpha)=-\sin(\alpha-\beta)$. Sine is an odd function, so $\sin(\beta-\alpha) = -\sin(\alpha-\beta)$.
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Identify the expression equivalent to $\cos(\beta-\alpha)$ in terms of $\cos(\alpha-\beta)$.
Identify the expression equivalent to $\cos(\beta-\alpha)$ in terms of $\cos(\alpha-\beta)$.
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$\cos(\beta-\alpha)=\cos(\alpha-\beta)$. Cosine is an even function, so $\cos(\beta-\alpha) = \cos(\alpha-\beta)$.
$\cos(\beta-\alpha)=\cos(\alpha-\beta)$. Cosine is an even function, so $\cos(\beta-\alpha) = \cos(\alpha-\beta)$.
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Identify the expression equivalent to $\tan(\beta-\alpha)$ in terms of $\tan(\alpha-\beta)$.
Identify the expression equivalent to $\tan(\beta-\alpha)$ in terms of $\tan(\alpha-\beta)$.
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$\tan(\beta-\alpha)=-\tan(\alpha-\beta)$. Tangent is an odd function, so $\tan(\beta-\alpha) = -\tan(\alpha-\beta)$.
$\tan(\beta-\alpha)=-\tan(\alpha-\beta)$. Tangent is an odd function, so $\tan(\beta-\alpha) = -\tan(\alpha-\beta)$.
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State the product-to-sum style result: what is $\sin\alpha\cos\beta+\cos\alpha\sin\beta$ equal to?
State the product-to-sum style result: what is $\sin\alpha\cos\beta+\cos\alpha\sin\beta$ equal to?
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$\sin\alpha\cos\beta+\cos\alpha\sin\beta=\sin(\alpha+\beta)$. This expression equals the sine addition formula result.
$\sin\alpha\cos\beta+\cos\alpha\sin\beta=\sin(\alpha+\beta)$. This expression equals the sine addition formula result.
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State the result: what is $\cos\alpha\cos\beta-\sin\alpha\sin\beta$ equal to?
State the result: what is $\cos\alpha\cos\beta-\sin\alpha\sin\beta$ equal to?
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$\cos\alpha\cos\beta-\sin\alpha\sin\beta=\cos(\alpha+\beta)$. This expression equals the cosine addition formula result.
$\cos\alpha\cos\beta-\sin\alpha\sin\beta=\cos(\alpha+\beta)$. This expression equals the cosine addition formula result.
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State the result: what is $\cos\alpha\cos\beta+\sin\alpha\sin\beta$ equal to?
State the result: what is $\cos\alpha\cos\beta+\sin\alpha\sin\beta$ equal to?
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$\cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\alpha-\beta)$. This expression equals the cosine subtraction formula result.
$\cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\alpha-\beta)$. This expression equals the cosine subtraction formula result.
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Identify the condition that makes $\tan(\alpha+\beta)$ undefined in $\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$.
Identify the condition that makes $\tan(\alpha+\beta)$ undefined in $\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$.
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$1-\tan\alpha\tan\beta=0$. When denominator equals zero, tangent is undefined.
$1-\tan\alpha\tan\beta=0$. When denominator equals zero, tangent is undefined.
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Identify the condition that makes $\tan(\alpha-\beta)$ undefined in $\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$.
Identify the condition that makes $\tan(\alpha-\beta)$ undefined in $\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$.
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$1+\tan\alpha\tan\beta=0$. When denominator equals zero, tangent is undefined.
$1+\tan\alpha\tan\beta=0$. When denominator equals zero, tangent is undefined.
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Identify the correct identity for $\sin(\alpha+\beta)$ among sign choices in the middle term.
Identify the correct identity for $\sin(\alpha+\beta)$ among sign choices in the middle term.
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$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. The correct sign in sine addition formula is positive.
$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. The correct sign in sine addition formula is positive.
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State the even-odd identity for sine: what is $\sin(-\theta)$?
State the even-odd identity for sine: what is $\sin(-\theta)$?
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$\sin(-\theta)=-\sin\theta$. Sine is an odd function: $\sin(-x) = -\sin(x)$.
$\sin(-\theta)=-\sin\theta$. Sine is an odd function: $\sin(-x) = -\sin(x)$.
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State the even-odd identity for cosine: what is $\cos(-\theta)$?
State the even-odd identity for cosine: what is $\cos(-\theta)$?
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$\cos(-\theta)=\cos\theta$. Cosine is an even function: $\cos(-x) = \cos(x)$.
$\cos(-\theta)=\cos\theta$. Cosine is an even function: $\cos(-x) = \cos(x)$.
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State the even-odd identity for tangent: what is $\tan(-\theta)$?
State the even-odd identity for tangent: what is $\tan(-\theta)$?
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$\tan(-\theta)=-\tan\theta$. Tangent is an odd function: $\tan(-x) = -\tan(x)$.
$\tan(-\theta)=-\tan\theta$. Tangent is an odd function: $\tan(-x) = -\tan(x)$.
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State the cofunction identity for sine: what is $\sin\left(\frac{\pi}{2}-\theta\right)$?
State the cofunction identity for sine: what is $\sin\left(\frac{\pi}{2}-\theta\right)$?
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$\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta$. Sine and cosine are cofunctions differing by $\frac{\pi}{2}$.
$\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta$. Sine and cosine are cofunctions differing by $\frac{\pi}{2}$.
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State the cofunction identity for cosine: what is $\cos\left(\frac{\pi}{2}-\theta\right)$?
State the cofunction identity for cosine: what is $\cos\left(\frac{\pi}{2}-\theta\right)$?
Tap to reveal answer
$\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta$. Cosine and sine are cofunctions differing by $\frac{\pi}{2}$.
$\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta$. Cosine and sine are cofunctions differing by $\frac{\pi}{2}$.
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Identify the expression equivalent to $\tan(\beta-\alpha)$ in terms of $\tan(\alpha-\beta)$.
Identify the expression equivalent to $\tan(\beta-\alpha)$ in terms of $\tan(\alpha-\beta)$.
Tap to reveal answer
$\tan(\beta-\alpha)=-\tan(\alpha-\beta)$. Tangent is an odd function, so $\tan(\beta-\alpha) = -\tan(\alpha-\beta)$.
$\tan(\beta-\alpha)=-\tan(\alpha-\beta)$. Tangent is an odd function, so $\tan(\beta-\alpha) = -\tan(\alpha-\beta)$.
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State the formula for $\sin(\alpha+\beta)$.
State the formula for $\sin(\alpha+\beta)$.
Tap to reveal answer
$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Standard angle addition formula for sine function.
$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Standard angle addition formula for sine function.
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State the formula for $\sin(\alpha-\beta)$.
State the formula for $\sin(\alpha-\beta)$.
Tap to reveal answer
$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$. Standard angle subtraction formula for sine function.
$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$. Standard angle subtraction formula for sine function.
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