Equivalent Representations of Trigonometric Functions - AP Precalculus
Card 1 of 30
Express $\cos(-\theta)$ using an identity.
Express $\cos(-\theta)$ using an identity.
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$\cos(-\theta) = \cos(\theta)$. Cosine is an even function, so $\cos(-x) = \cos(x)$.
$\cos(-\theta) = \cos(\theta)$. Cosine is an even function, so $\cos(-x) = \cos(x)$.
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Find the value of $\sec(0)$.
Find the value of $\sec(0)$.
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$\sec(0) = 1$. Since $\cos(0) = 1$, its reciprocal is 1.
$\sec(0) = 1$. Since $\cos(0) = 1$, its reciprocal is 1.
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Express $\cos(2\theta)$ using a double angle identity.
Express $\cos(2\theta)$ using a double angle identity.
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$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$. Double angle identity for cosine function.
$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$. Double angle identity for cosine function.
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Express $\tan(2\theta)$ using a double angle identity.
Express $\tan(2\theta)$ using a double angle identity.
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$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$. Double angle identity for tangent function.
$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$. Double angle identity for tangent function.
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Express $\sin(\frac{\pi}{2} - \theta)$ using a co-function identity.
Express $\sin(\frac{\pi}{2} - \theta)$ using a co-function identity.
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$\sin(\frac{\pi}{2} - \theta) = \cos(\theta)$. Co-function identity: sine of complementary angle equals cosine.
$\sin(\frac{\pi}{2} - \theta) = \cos(\theta)$. Co-function identity: sine of complementary angle equals cosine.
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Identify the period of $\tan(\theta)$.
Identify the period of $\tan(\theta)$.
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The period of $\tan(\theta)$ is $\pi$. Tangent completes one cycle every $\pi$ radians.
The period of $\tan(\theta)$ is $\pi$. Tangent completes one cycle every $\pi$ radians.
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Express $\tan(-\theta)$ using an identity.
Express $\tan(-\theta)$ using an identity.
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$\tan(-\theta) = -\tan(\theta)$. Tangent is an odd function, so $\tan(-x) = -\tan(x)$.
$\tan(-\theta) = -\tan(\theta)$. Tangent is an odd function, so $\tan(-x) = -\tan(x)$.
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Convert radians to degrees for $\frac{\pi}{3}$.
Convert radians to degrees for $\frac{\pi}{3}$.
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$\frac{\pi}{3}$ radians = $60^\circ$. Multiply by $\frac{180}{\pi}$ to convert radians to degrees.
$\frac{\pi}{3}$ radians = $60^\circ$. Multiply by $\frac{180}{\pi}$ to convert radians to degrees.
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Convert degrees to radians for $180^\circ$.
Convert degrees to radians for $180^\circ$.
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$180^\circ$ = $\pi$ radians. Multiply by $\frac{\pi}{180}$ to convert degrees to radians.
$180^\circ$ = $\pi$ radians. Multiply by $\frac{\pi}{180}$ to convert degrees to radians.
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Identify the period of $\sin(\theta)$.
Identify the period of $\sin(\theta)$.
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The period of $\sin(\theta)$ is $2\pi$. Sine completes one cycle every $2\pi$ radians.
The period of $\sin(\theta)$ is $2\pi$. Sine completes one cycle every $2\pi$ radians.
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Identify the period of $\tan(\theta)$.
Identify the period of $\tan(\theta)$.
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The period of $\tan(\theta)$ is $\pi$. Tangent completes one cycle every $\pi$ radians.
The period of $\tan(\theta)$ is $\pi$. Tangent completes one cycle every $\pi$ radians.
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Which quadrant is $\cos(\theta) > 0$ and $\sin(\theta) < 0$?
Which quadrant is $\cos(\theta) > 0$ and $\sin(\theta) < 0$?
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Quadrant IV. In Quadrant IV, cosine is positive and sine is negative.
Quadrant IV. In Quadrant IV, cosine is positive and sine is negative.
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Find the value of $\tan(\frac{\pi}{4})$.
Find the value of $\tan(\frac{\pi}{4})$.
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$\tan(\frac{\pi}{4}) = 1$. At 45°, tangent equals 1 since opposite equals adjacent.
$\tan(\frac{\pi}{4}) = 1$. At 45°, tangent equals 1 since opposite equals adjacent.
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Find the value of $\csc(\frac{\pi}{2})$.
Find the value of $\csc(\frac{\pi}{2})$.
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$\csc(\frac{\pi}{2}) = 1$. Since $\sin(\frac{\pi}{2}) = 1$, its reciprocal is 1.
$\csc(\frac{\pi}{2}) = 1$. Since $\sin(\frac{\pi}{2}) = 1$, its reciprocal is 1.
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What is the sine of an angle expressed as a ratio in a right triangle?
What is the sine of an angle expressed as a ratio in a right triangle?
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Sine = $\frac{\text{opposite}}{\text{hypotenuse}}$. Basic right triangle definition for sine function.
Sine = $\frac{\text{opposite}}{\text{hypotenuse}}$. Basic right triangle definition for sine function.
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What is the cosine of an angle expressed as a ratio in a right triangle?
What is the cosine of an angle expressed as a ratio in a right triangle?
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Cosine = $\frac{\text{adjacent}}{\text{hypotenuse}}$. Basic right triangle definition for cosine function.
Cosine = $\frac{\text{adjacent}}{\text{hypotenuse}}$. Basic right triangle definition for cosine function.
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What is the tangent of an angle expressed as a ratio in a right triangle?
What is the tangent of an angle expressed as a ratio in a right triangle?
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Tangent = $\frac{\text{opposite}}{\text{adjacent}}$. Basic right triangle definition for tangent function.
Tangent = $\frac{\text{opposite}}{\text{adjacent}}$. Basic right triangle definition for tangent function.
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Find the value of $\sin(0)$.
Find the value of $\sin(0)$.
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$\sin(0) = 0$. At zero radians, sine equals zero.
$\sin(0) = 0$. At zero radians, sine equals zero.
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What is the reciprocal identity for secant?
What is the reciprocal identity for secant?
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$\sec(\theta) = \frac{1}{\cos(\theta)}$. Secant is the reciprocal of cosine.
$\sec(\theta) = \frac{1}{\cos(\theta)}$. Secant is the reciprocal of cosine.
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Find the value of $\cot(\frac{\pi}{4})$.
Find the value of $\cot(\frac{\pi}{4})$.
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$\cot(\frac{\pi}{4}) = 1$. Since $\tan(\frac{\pi}{4}) = 1$, its reciprocal is 1.
$\cot(\frac{\pi}{4}) = 1$. Since $\tan(\frac{\pi}{4}) = 1$, its reciprocal is 1.
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Express $\sin(2\theta)$ using a double angle identity.
Express $\sin(2\theta)$ using a double angle identity.
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$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. Double angle identity for sine function.
$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. Double angle identity for sine function.
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Express $\cos(\frac{\pi}{2} - \theta)$ using a co-function identity.
Express $\cos(\frac{\pi}{2} - \theta)$ using a co-function identity.
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$\cos(\frac{\pi}{2} - \theta) = \sin(\theta)$. Co-function identity: cosine of complementary angle equals sine.
$\cos(\frac{\pi}{2} - \theta) = \sin(\theta)$. Co-function identity: cosine of complementary angle equals sine.
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Express $\tan(\frac{\pi}{2} - \theta)$ using a co-function identity.
Express $\tan(\frac{\pi}{2} - \theta)$ using a co-function identity.
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$\tan(\frac{\pi}{2} - \theta) = \cot(\theta)$. Co-function identity: tangent of complementary angle equals cotangent.
$\tan(\frac{\pi}{2} - \theta) = \cot(\theta)$. Co-function identity: tangent of complementary angle equals cotangent.
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Express $\sin(\theta + \pi)$ using an identity.
Express $\sin(\theta + \pi)$ using an identity.
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$\sin(\theta + \pi) = -\sin(\theta)$. Adding $\pi$ to angle negates sine value.
$\sin(\theta + \pi) = -\sin(\theta)$. Adding $\pi$ to angle negates sine value.
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Express $\cos(\theta + \pi)$ using an identity.
Express $\cos(\theta + \pi)$ using an identity.
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$\cos(\theta + \pi) = -\cos(\theta)$. Adding $\pi$ to angle negates cosine value.
$\cos(\theta + \pi) = -\cos(\theta)$. Adding $\pi$ to angle negates cosine value.
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Which quadrant is $\sin(\theta) > 0$ and $\cos(\theta) < 0$?
Which quadrant is $\sin(\theta) > 0$ and $\cos(\theta) < 0$?
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Quadrant II. In Quadrant II, sine is positive and cosine is negative.
Quadrant II. In Quadrant II, sine is positive and cosine is negative.
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Express sine in terms of cosine for an angle $\theta$.
Express sine in terms of cosine for an angle $\theta$.
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$\sin(\theta) = \cos(\frac{\pi}{2} - \theta)$. Co-function identity relating sine and cosine.
$\sin(\theta) = \cos(\frac{\pi}{2} - \theta)$. Co-function identity relating sine and cosine.
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Express cosine in terms of sine for an angle $\theta$.
Express cosine in terms of sine for an angle $\theta$.
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$\cos(\theta) = \sin(\frac{\pi}{2} - \theta)$. Co-function identity relating cosine and sine.
$\cos(\theta) = \sin(\frac{\pi}{2} - \theta)$. Co-function identity relating cosine and sine.
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Express tangent in terms of sine and cosine for an angle $\theta$.
Express tangent in terms of sine and cosine for an angle $\theta$.
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$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Quotient identity expressing tangent in terms of sine and cosine.
$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Quotient identity expressing tangent in terms of sine and cosine.
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What is the identity for $\sin^2(\theta) + \cos^2(\theta)$?
What is the identity for $\sin^2(\theta) + \cos^2(\theta)$?
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$\sin^2(\theta) + \cos^2(\theta) = 1$. Pythagorean identity, fundamental trigonometric relationship.
$\sin^2(\theta) + \cos^2(\theta) = 1$. Pythagorean identity, fundamental trigonometric relationship.
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