The Tangent Function - AP Precalculus
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What is the tangent of 45 degrees?
What is the tangent of 45 degrees?
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- At $45°$, opposite and adjacent sides are equal in length.
- At $45°$, opposite and adjacent sides are equal in length.
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Find the tangent of $\frac{\pi}{6}$ radians.
Find the tangent of $\frac{\pi}{6}$ radians.
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$\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$. In a 30-60-90 triangle, $\tan(30°) = \frac{1}{\sqrt{3}}$.
$\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$. In a 30-60-90 triangle, $\tan(30°) = \frac{1}{\sqrt{3}}$.
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State the formula for the tangent of an angle $\theta$ in a right triangle.
State the formula for the tangent of an angle $\theta$ in a right triangle.
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$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Basic trigonometric ratio formula using sides of a right triangle.
$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Basic trigonometric ratio formula using sides of a right triangle.
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Find the tangent of $\frac{4\pi}{3}$ radians.
Find the tangent of $\frac{4\pi}{3}$ radians.
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$\sqrt{3}$. Reference angle is $\frac{\pi}{3}$ in quadrant III.
$\sqrt{3}$. Reference angle is $\frac{\pi}{3}$ in quadrant III.
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What is the tangent of 60 degrees?
What is the tangent of 60 degrees?
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$\sqrt{3}$. Standard value from the 30-60-90 triangle.
$\sqrt{3}$. Standard value from the 30-60-90 triangle.
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What is the tangent of 90 degrees?
What is the tangent of 90 degrees?
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Undefined. At $90°$, adjacent side is 0, making division undefined.
Undefined. At $90°$, adjacent side is 0, making division undefined.
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What is the range of the tangent function?
What is the range of the tangent function?
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All real numbers. Tangent can take any real value from $-\infty$ to $+\infty$.
All real numbers. Tangent can take any real value from $-\infty$ to $+\infty$.
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What is the period of the tangent function?
What is the period of the tangent function?
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$\pi$. Tangent repeats its values every $\pi$ radians.
$\pi$. Tangent repeats its values every $\pi$ radians.
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Identify the vertical asymptotes of the tangent function.
Identify the vertical asymptotes of the tangent function.
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$x = \frac{\pi}{2} + n\pi$, $n \in \mathbb{Z}$. Occurs where cosine equals zero, causing division by zero.
$x = \frac{\pi}{2} + n\pi$, $n \in \mathbb{Z}$. Occurs where cosine equals zero, causing division by zero.
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Find the tangent of $\frac{\pi}{4}$ radians.
Find the tangent of $\frac{\pi}{4}$ radians.
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- At $45°$, opposite and adjacent sides are equal.
- At $45°$, opposite and adjacent sides are equal.
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Find the tangent of $\frac{\pi}{6}$ radians.
Find the tangent of $\frac{\pi}{6}$ radians.
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$\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$. In a 30-60-90 triangle, $\tan(30°) = \frac{1}{\sqrt{3}}$.
$\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$. In a 30-60-90 triangle, $\tan(30°) = \frac{1}{\sqrt{3}}$.
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Find the tangent of $\frac{\pi}{3}$ radians.
Find the tangent of $\frac{\pi}{3}$ radians.
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$\sqrt{3}$. In a 30-60-90 triangle, $\tan(60°) = \sqrt{3}$.
$\sqrt{3}$. In a 30-60-90 triangle, $\tan(60°) = \sqrt{3}$.
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What is the tangent of 180 degrees?
What is the tangent of 180 degrees?
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- At $180°$, we're on the negative x-axis where tangent is 0.
- At $180°$, we're on the negative x-axis where tangent is 0.
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What is the definition of the tangent function in a right triangle?
What is the definition of the tangent function in a right triangle?
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Tangent is the ratio of the opposite side to the adjacent side. Fundamental definition for the tangent ratio in right triangles.
Tangent is the ratio of the opposite side to the adjacent side. Fundamental definition for the tangent ratio in right triangles.
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State the formula for the tangent of an angle $\theta$ in a right triangle.
State the formula for the tangent of an angle $\theta$ in a right triangle.
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$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Basic trigonometric ratio formula using sides of a right triangle.
$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Basic trigonometric ratio formula using sides of a right triangle.
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What is the tangent of 0 degrees?
What is the tangent of 0 degrees?
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- At $0°$, opposite side is 0 while adjacent side is positive.
- At $0°$, opposite side is 0 while adjacent side is positive.
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Find the tangent of $\frac{5\pi}{6}$ radians.
Find the tangent of $\frac{5\pi}{6}$ radians.
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$-\frac{1}{\sqrt{3}}$ or $-\frac{\sqrt{3}}{3}$. At $150°$, we're in quadrant II where tangent is negative.
$-\frac{1}{\sqrt{3}}$ or $-\frac{\sqrt{3}}{3}$. At $150°$, we're in quadrant II where tangent is negative.
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Find the tangent of $270$ degrees.
Find the tangent of $270$ degrees.
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Undefined. At $270°$, cosine is 0, making tangent undefined.
Undefined. At $270°$, cosine is 0, making tangent undefined.
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Find the value of $\tan(\pi)$.
Find the value of $\tan(\pi)$.
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- $\pi$ radians equals $180°$ where tangent is zero.
- $\pi$ radians equals $180°$ where tangent is zero.
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Find the tangent of $\frac{3\pi}{4}$ radians.
Find the tangent of $\frac{3\pi}{4}$ radians.
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-1. At $135°$, we're in quadrant II where tangent is negative.
-1. At $135°$, we're in quadrant II where tangent is negative.
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What is the tangent of $2\pi$ radians?
What is the tangent of $2\pi$ radians?
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- $2\pi$ radians equals $360°$, same as $0°$.
- $2\pi$ radians equals $360°$, same as $0°$.
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What is the fundamental period of the tangent function?
What is the fundamental period of the tangent function?
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$\pi$. The smallest positive period for the tangent function.
$\pi$. The smallest positive period for the tangent function.
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Identify the symmetry type of the tangent function.
Identify the symmetry type of the tangent function.
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Odd function. Since $\tan(-x) = -\tan(x)$, tangent is an odd function.
Odd function. Since $\tan(-x) = -\tan(x)$, tangent is an odd function.
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What is the derivative of $\tan(x)$?
What is the derivative of $\tan(x)$?
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$\sec^2(x)$. Standard derivative formula for the tangent function.
$\sec^2(x)$. Standard derivative formula for the tangent function.
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What is the integral of $\tan(x)$?
What is the integral of $\tan(x)$?
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$-\ln|\cos(x)| + C$. Standard antiderivative formula for the tangent function.
$-\ln|\cos(x)| + C$. Standard antiderivative formula for the tangent function.
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Determine the value of $\tan(\frac{\pi}{2} - x)$.
Determine the value of $\tan(\frac{\pi}{2} - x)$.
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$\cot(x)$. Using the cofunction identity for complementary angles.
$\cot(x)$. Using the cofunction identity for complementary angles.
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What is $\tan(x)$ in terms of sine and cosine?
What is $\tan(x)$ in terms of sine and cosine?
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$\tan(x) = \frac{\sin(x)}{\cos(x)}$. Fundamental relationship between tangent, sine, and cosine.
$\tan(x) = \frac{\sin(x)}{\cos(x)}$. Fundamental relationship between tangent, sine, and cosine.
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Find $\tan(\pi + x)$ in terms of $\tan(x)$.
Find $\tan(\pi + x)$ in terms of $\tan(x)$.
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$\tan(x)$. Tangent has period $\pi$, so adding $\pi$ doesn't change the value.
$\tan(x)$. Tangent has period $\pi$, so adding $\pi$ doesn't change the value.
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What is the tangent of 45 degrees?
What is the tangent of 45 degrees?
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- At $45°$, opposite and adjacent sides are equal in length.
- At $45°$, opposite and adjacent sides are equal in length.
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What is the tangent of 60 degrees?
What is the tangent of 60 degrees?
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$\sqrt{3}$. Standard value from the 30-60-90 triangle.
$\sqrt{3}$. Standard value from the 30-60-90 triangle.
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