AP Calculus Unit Circle Flashcards Flashcards

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Unit circle basics

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The **unit circle** is the circle of radius 1 centered at the origin. Every point on it has coordinates $(\cos \theta, \sin \theta)$ for an angle $\theta$ in **standard position**. Because the radius is 1, any point satisfies $x^2 + y^2 = 1$.

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0° angle

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At $0°$ (or $0$ radians), the coordinates are $(1, 0)$, so $\cos 0° = 1$ and $\sin 0° = 0$. The tangent is $\tan 0° = 0$. This point lies on the positive x-axis.

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90° angle

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At $90°$ ( $\pi/2$ ), the coordinates are $(0, 1)$. So $\cos 90° = 0$ and $\sin 90° = 1$. Tangent is undefined because $\cos 90° = 0$.

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180° angle

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At $180°$ ( $\pi$ ), the coordinates are $(-1, 0)$. So $\cos 180° = -1$ and $\sin 180° = 0$. Tangent is $0$ because $\tan 180° = \sin 180°/\cos 180° = 0/(-1) = 0$.

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270° angle

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At $270°$ ( $\tfrac{3\pi}{2}$ ), the coordinates are $(0, -1)$. So $\cos 270° = 0$ and $\sin 270° = -1$. Tangent is undefined because $\cos 270° = 0$.

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Cosine values

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Cosine is the **x-coordinate** of a unit circle point: $\cos \theta = x$. Key values include $\cos 0° = 1$, $\cos 30° = $ $\frac{\sqrt{3}}{2}$, $\cos 45° = $ $\frac{\sqrt{2}}{2}$, $\cos 60° = $ $\frac{1}{2}$, and $\cos 90° = 0$. Use symmetry to find cosine values in other quadrants.

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Sine values

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Sine is the **y-coordinate**: $\sin \theta = y$. Examples: $\sin 0° = 0$, $\sin 30° = $ $\frac{1}{2}$, $\sin 45° = $ $\frac{\sqrt{2}}{2}$, $\sin 60° = $ $\frac{\sqrt{3}}{2}$, and $\sin 90° = 1$. Values extend to other angles via symmetry in the unit circle.

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Tangent values

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Tangent is the ratio $\tan \theta = \dfrac{\sin \theta}{\cos \theta}$ (where defined). Examples: $\tan 0° = 0$, $\tan 45° = 1$, $\tan 60° = $ $\sqrt{3}$, and $\tan 90°$ is undefined because $\cos 90° = 0$. Tangent repeats with period $2\pi$ when defined.

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Quadrant signs

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Sign patterns: In **QI** both sine and cosine are positive; **QII** sine positive, cosine negative; **QIII** both negative; **QIV** cosine positive, sine negative. These signs determine the signs of sine and cosine for angle values.

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Pythagorean identity

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The **Pythagorean identity** is $\sin^2 \theta + \cos^2 \theta = 1$ for all $\theta$. This follows from substituting $x = \cos \theta$, $y = \sin \theta$ into $x^2 + y^2 = 1$ on the unit circle.

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Reference angles

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A **reference angle** is the acute angle formed with the x-axis. Its sine and cosine magnitudes equal those of the corresponding first-quadrant angle, with signs determined by the quadrant. Use reference angles to simplify computing sine and cosine for any angle.

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Radians labels

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Angles on the unit circle are often given in radians; common values include $0$, $\tfrac{\pi}{6}$, $\tfrac{\pi}{4}$, $\tfrac{\pi}{3}$, $\tfrac{\pi}{2}$. Other key marks are $\tfrac{2\pi}{3}$, $\tfrac{3\pi}{4}$, $\tfrac{5\pi}{6}$, and $\pi$. Remember that $180° = \pi$ radians.

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Key angles list

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Key angles on the unit circle include $0°$, $30°$, $45°$, $60°$, $90°$, $120°$, $135°$, $150°$, $180°$, $210°$, $225°$, $240°$, $270°$, $300°$, $315°$, $330°$. Each angle has a corresponding $(\cos \theta, \sin \theta)$ value, often in radical form.

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Coordinate pairs

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The coordinates on the circle are $(\cos \theta, \sin \theta)$. For standard angles they are written as radical values like $(1,0)$, $(\tfrac{\sqrt{3}}{2}, \tfrac{1}{2})$, $(0,1)$, $( -\tfrac{\sqrt{3}}{2}, \tfrac{1}{2})$, etc.

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Sine vs Cosine

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**Sine vs Cosine**: $\sin \theta$ gives the vertical coordinate and $\cos \theta$ the horizontal. Their squares sum to 1: $\sin^2 \theta + \cos^2 \theta = 1$.

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Unit circle equation

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**Unit circle equation**: $x^2 + y^2 = 1$ for all points on the circle of radius 1. Substituting $x=\cos \theta$, $y=\sin \theta$ yields the Pythagorean identity.

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Symmetry properties

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The unit circle has symmetry across the axes; coordinates reflect across quadrants. Sine and cosine signs follow quadrant rules; for example, $\sin(-\theta) = -\sin \theta$ and $\cos(-\theta) = \cos \theta$.

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Angles in radians list

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Common radian measures include $0$, $\tfrac{\pi}{6}$, $\tfrac{\pi}{4}$, $\tfrac{\pi}{3}$, $\tfrac{\pi}{2}$, $\tfrac{2\pi}{3}$, $\tfrac{3\pi}{4}$, $\tfrac{5\pi}{6}$, $\pi$, $\tfrac{7\pi}{6}$, $\tfrac{5\pi}{4}$, $\tfrac{4\pi}{3}$, $\tfrac{3\pi}{2}$, $\tfrac{5\pi}{3}$, $\tfrac{7\pi}{3}$, $2\pi$. These correspond to the same degree measures and provide the same cosine and sine values.

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Periodicity

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**Periodicity**: Sine and cosine have period $2\pi$; thus $\sin(\theta+2\pi) = \sin \theta$ and $\cos(\theta+2\pi) = \cos \theta$. This means the unit circle values repeat every full revolution.