Unit Circle Essentials Flashcards

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Unit circle

Back

A circle of radius 1 centered at the origin. For any angle $\theta$, the point on the circle is $(\cos \theta, \sin \theta)$, so $\cos \theta$ is the x-coordinate and $\sin \theta$ is the y-coordinate.

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Cos θ

Back

Cosine is the x-coordinate on the unit circle. Specifically, $\cos \theta$ gives the horizontal position and ranges between $-1$ and $1$; it equals $1$ at $\theta = 0$ and $0$ at $\theta = \frac{\pi}{2}$.

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Sin θ

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Sine is the y-coordinate on the unit circle. $\sin \theta$ is the vertical position and ranges between $-1$ and $1$; it equals $0$ at $\theta = 0, \pi, 2\pi$.

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Tan θ

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Tangent on the unit circle is $\tan \theta = \frac{\sin \theta}{\cos \theta}$ wherever $\cos \theta \neq 0$. It represents the slope of the line through the origin corresponding to angle $\theta$.

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Reference angle

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A reference angle is the acute angle formed with the x-axis in a given quadrant. The sine and cosine values are the same as those at the reference angle, but signs are determined by the quadrant.

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Special angle 0

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At $\theta = 0$, $\cos 0 = 1$, $\sin 0 = 0$, and $\tan 0 = 0$. The corresponding point on the unit circle is $(1,0)$.

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Special angle π/6

Back

For $\theta = \frac{\pi}{6}$, $(\cos \frac{\pi}{6}, \sin \frac{\pi}{6}) = (\frac{\sqrt{3}}{2}, \frac{1}{2})$. These values are anchor points used to derive other angles and values on the unit circle.

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Special angle π/4

Back

For $\theta = \frac{\pi}{4}$, $(\cos \frac{\pi}{4}, \sin \frac{\pi}{4}) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$. In Quadrant I these are equal, reflecting symmetry.

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Special angle π/3

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For $\theta = \frac{\pi}{3}$, $(\cos \frac{\pi}{3}, \sin \frac{\pi}{3}) = (\frac{1}{2}, \frac{\sqrt{3}}{2})$.

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Special angle π/2

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For $\theta = \frac{\pi}{2}$, $(\cos \frac{\pi}{2}, \sin \frac{\pi}{2}) = (0, 1)$. This is the top point on the unit circle.

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Angles in radians

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The unit circle uses radians; a full rotation is $2\pi$ radians. Common reference angles include $0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$.

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Quadrant signs

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The signs of sine and cosine depend on the quadrant. I: both positive; II: sine positive and cosine negative; III: both negative; IV: sine negative and cosine positive. Tangent follows the sign of the ratio $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

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Pythagorean identity

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On the unit circle, $\sin^2 \theta + \cos^2 \theta = 1$. This reflects the radius being 1 and the Pythagorean theorem.

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Reciprocal identities

Back

For the unit circle, $\sec \theta = \frac{1}{\cos \theta}$, $\csc \theta = \frac{1}{\sin \theta}$, and $\cot \theta = \frac{\cos \theta}{\sin \theta}$, defined where the denominators are nonzero.

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Cofunctions

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Cofunction identities relate sine and cosine at complementary angles: $\sin\bigl(\frac{\pi}{2}-\theta\bigr) = \cos \theta$ and $\cos\bigl(\frac{\pi}{2}-\theta\bigr) = \sin \theta$.

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Sine signs

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Sine is positive in Quadrants I and II, and negative in Quadrants III and IV. This follows from the y-coordinate on the unit circle.

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Cosine signs

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Cosine is positive in Quadrants I and IV, and negative in Quadrants II and III. This follows from the x-coordinate on the unit circle.

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Cos θ = 0

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Cosine equals zero at $\theta = \frac{\pi}{2}$ and $\frac{3\pi}{2}$ (mod $2\pi$). The corresponding points are $(0,1)$ and $(0,-1)$ on the unit circle.

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Sin θ = 0

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Sine equals zero at $\theta = 0, \pi, 2\pi$ (mod $2\pi$). The corresponding points are $(1,0)$ and $(-1,0)$.

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Periodicity

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Trig functions on the unit circle are periodic with period $2\pi$: $\sin(\theta + 2\pi) = \sin \theta$ and $\cos(\theta + 2\pi) = \cos \theta$. Therefore angles differing by $2\pi$ share the same coordinates on the circle.