Unit Circle Mechanics and Trigonometric Coordinates

Core Concepts

Understand coordinates on the unit circle, circular definitions, signs, and reference angles.

Unit Circle Mechanics and Trigonometric Coordinates

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What is the equation of the unit circle centered at the origin?

$$x^2 + y^2 = 1$$ It has a radius $r = 1$ centered at $(0, 0)$.

How is $\tan\theta$ defined in terms of the unit circle coordinates $(x, y)$?

$$\tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta} \quad (x \neq 0)$$ $\tan\theta$ represents the slope of the terminal ray.

How does the unit circle equation $x^2 + y^2 = 1$ establish the Pythagorean trigonometric identity?

Substituting $x = \cos\theta$ and $y = \sin\theta$ into $x^2 + y^2 = 1$ yields: $$\cos^2\theta + \sin^2\theta = 1$$

What is the reference angle $\theta'$ (or $\theta_r$) for an angle $\theta$ in standard position?

Diagram of an angle theta in standard position terminating in quadrant two, showing its reference angle theta prime formed between the terminal ray and the negative x-axis on a coordinate plane with a circle centered at the origin.

The acute positive angle formed between the terminal side of $\theta$ and the horizontal $x$-axis. Reference angles always satisfy $0 \le \theta' \le \frac{\pi}{2}$.

Find the reference angle for $\theta = \frac{5\pi}{6}$.

Since $\frac{5\pi}{6}$ is in Quadrant II: $$\theta' = \pi - \frac{5\pi}{6} = \frac{\pi}{6}$$

Find the reference angle for $\theta = \frac{4\pi}{3}$.

Since $\frac{4\pi}{3}$ is in Quadrant III: $$\theta' = \frac{4\pi}{3} - \pi = \frac{\pi}{3}$$

What mnemonic identifies which trigonometric functions are positive in Quadrants I through IV?

A Cartesian plane illustrating the ASTC rule across four quadrants: Quadrant I has all positive, Quadrant II has sine positive, Quadrant III has tangent positive, and Quadrant IV has cosine positive.

'All Students Take Calculus' (ASTC): Quadrant I: All positive Quadrant II: Sine positive Quadrant III: Tangent positive Quadrant IV: Cosine positive

What are the unit circle coordinates and trig values for $\theta = 0$?

Coordinates: $(1, 0)$\n$$\cos 0 = 1$$\n$$\sin 0 = 0$$\n$$\tan 0 = 0$$

What are the unit circle coordinates and trig values for $\theta = \frac{\pi}{2}$?

Coordinates: $(0, 1)$\n$$\cos\frac{\pi}{2} = 0$$\n$$\sin\frac{\pi}{2} = 1$$\n$$\tan\frac{\pi}{2} = \text{undefined}$$

What are the unit circle coordinates and trig values for $\theta = \pi$?

Coordinates: $(-1, 0)$\n$$\cos\pi = -1$$\n$$\sin\pi = 0$$\n$$\tan\pi = 0$$

What are the unit circle coordinates and trig values for $\theta = \frac{3\pi}{2}$?

Coordinates: $(0, -1)$\n$$\cos\frac{3\pi}{2} = 0$$\n$$\sin\frac{3\pi}{2} = -1$$\n$$\tan\frac{3\pi}{2} = \text{undefined}$$

Why are $\tan\theta$ and $\sec\theta$ undefined at $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$?

Both have $\cos\theta$ (or $x$) in the denominator: $$\tan\theta = \frac{y}{x}, \quad \sec\theta = \frac{1}{x}$$ At these angles, $x = 0$, resulting in division by zero.

For a point $P(x, y)$ where the terminal side of angle $\theta$ intersects the unit circle, how are $\cos\theta$ and $\sin\theta$ defined?

Unit circle on a coordinate grid showing a right triangle in the first quadrant, where an angle theta gives horizontal side cosine of theta, vertical side sine of theta, hypotenuse 1, and terminal point P with coordinates cosine of theta, sine of theta.

$$x = \cos\theta$$ $$y = \sin\theta$$ Thus, any point on the unit circle has coordinates $(\cos\theta, \sin\theta)$.

How do you calculate the reference angle $\theta'$ in each of Quadrants II, III, and IV (in radians)?

A coordinate plane diagram illustrating formulas for finding the reference angle theta in Quadrants 2, 3, and 4 as pi minus theta, theta minus pi, and 2 pi minus theta, respectively, measured relative to the x-axis.

Quadrant II: $$\theta' = \pi - \theta$$ Quadrant III: $$\theta' = \theta - \pi$$ Quadrant IV: $$\theta' = 2\pi - \theta$$

Unit Circle Mechanics and Trigonometric Coordinates

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