Chapter 7 — The Unit Circle: Sine and Cosine Functions

An angle is in standard position when:

• Its vertex is at the origin.
• Its initial side lies along the positive $x$-axis.
• Its terminal side is the ray after rotation.

Positive angles: counterclockwise rotation.
Negative angles: clockwise rotation.

One full rotation = $360°$

Common angle types:

• Acute: $0° < \theta < 90°$
• Right: $\theta = 90°$
• Obtuse: $90° < \theta < 180°$
• Straight: $\theta = 180°$
• Reflex: $180° < \theta < 360°$

Degrees-Minutes-Seconds (DMS): $1° = 60’$ (minutes), $1’ = 60’’$ (seconds).

One radian is the angle subtended at the center of a circle by an arc whose length equals the radius.

\(2\pi \text{ radians} = 360° \qquad \Rightarrow \qquad \pi \text{ rad} = 180°\)

\(\text{Radians} \to \text{Degrees:} \quad \theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}\)

Two angles are coterminal if they share the same terminal side.

\(\theta_{\text{coterminal}} = \theta + 360°n \quad \text{(degrees)}\) \(\theta_{\text{coterminal}} = \theta + 2\pi n \quad \text{(radians)}\)

where $n$ is any integer.

For a circle of radius $r$ and central angle $\theta$ (in radians):

\(\text{Sector area:} \quad A = \frac{1}{2}r^2\theta\)

• Angular speed: $\omega = \dfrac{\theta}{t}$ (radians per unit time)
• Linear speed: $v = \dfrac{s}{t} = r\omega$ (distance per unit time along the arc)

Relationship: $v = r\omega$

For a right triangle with angle $\theta$, opposite side ($O$), adjacent side ($A$), and hypotenuse ($H$):

| Function | Ratio | Mnemonic | |———-|——-|———-| | $\sin\theta$ | $\dfrac{O}{H}$ | SOH | | $\cos\theta$ | $\dfrac{A}{H}$ | CAH | | $\tan\theta$ | $\dfrac{O}{A}$ | TOA | | $\csc\theta$ | $\dfrac{H}{O}$ | reciprocal of sine | | $\sec\theta$ | $\dfrac{H}{A}$ | reciprocal of cosine | | $\cot\theta$ | $\dfrac{A}{O}$ | reciprocal of tangent |

45-45-90 Triangle: Sides in ratio $1 : 1 : \sqrt{2}$

30-60-90 Triangle: Sides in ratio $1 : \sqrt{3} : 2$

| Angle | sin | cos | tan | |——-|—–|—–|—–| | $30°$ | $\dfrac{1}{2}$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$ | | $60°$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{1}{2}$ | $\sqrt{3}$ |

Cofunction Identities: If $\alpha + \beta = 90°$ (or $\alpha + \beta = \tfrac{\pi}{2}$), then:

\(\sin\alpha = \cos\beta, \qquad \cos\alpha = \sin\beta\) \(\tan\alpha = \cot\beta, \qquad \sec\alpha = \csc\beta\)

In other words:

\(\sin(90° - \theta) = \cos\theta, \qquad \cos(90° - \theta) = \sin\theta\)

The unit circle is the circle of radius 1 centered at the origin:

Every point on the unit circle can be written as $(\cos\theta, \sin\theta)$ where $\theta$ is the angle from the positive $x$-axis.

For any angle $\theta$, the terminal side intersects the unit circle at a point $(x, y)$. Then:

This definition extends sine and cosine to all real numbers, not just acute angles.

The reference angle $\theta’$ for an angle $\theta$ in standard position is the acute angle formed between the terminal side and the $x$-axis.

| Quadrant | Reference angle formula | |———-|———————-| | I ($0° < \theta < 90°$) | $\theta’ = \theta$ | | II ($90° < \theta < 180°$) | $\theta’ = 180° - \theta$ | | III ($180° < \theta < 270°$) | $\theta’ = \theta - 180°$ | | IV ($270° < \theta < 360°$) | $\theta’ = 360° - \theta$ |

“All Students Take Calculus” (ASTC):

| Quadrant | sin | cos | tan | |———-|—–|—–|—–| | I | + | + | + | (All positive) | II | + | − | − | (Sine positive) | III | − | − | + | (Tangent positive) | IV | − | + | − | (Cosine positive)

| Function | Definition | Domain restriction | |———-|———–|——————-| | $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ | $= \dfrac{y}{x}$ on unit circle | $\cos\theta \ne 0$ ($\theta \ne 90° + 180°n$) | | $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$ | $= \dfrac{x}{y}$ on unit circle | $\sin\theta \ne 0$ ($\theta \ne 180°n$) | | $\sec\theta = \dfrac{1}{\cos\theta}$ | $= \dfrac{1}{x}$ on unit circle | $\cos\theta \ne 0$ | | $\csc\theta = \dfrac{1}{\sin\theta}$ | $= \dfrac{1}{y}$ on unit circle | $\sin\theta \ne 0$ |

Reciprocal Identities:

Quotient Identities:

Pythagorean Identities:

\(\sin^2\theta + \cos^2\theta = 1\) \(1 + \tan^2\theta = \sec^2\theta\) \(1 + \cot^2\theta = \csc^2\theta\)

Even/Odd Identities:

\(\sin(-\theta) = -\sin\theta \quad \text{(odd)}\) \(\cos(-\theta) = \cos\theta \quad \text{(even)}\) \(\tan(-\theta) = -\tan\theta \quad \text{(odd)}\)