5.1 Angles

Angles and Trigonometry

Angles in standard position are the starting point for everything in trigonometry. By placing angles on a coordinate plane with a consistent setup, you get a precise way to describe rotation, position, and eventually the trig functions themselves.

Degrees and radians are two systems for measuring angles, and you'll need to move between them fluently. This section also covers coterminal angles, arc length, and speed of rotation, all of which build directly on these angle fundamentals.

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Angles in Standard Position

An angle is in standard position when its vertex sits at the origin (0, 0) and its initial side lies along the positive x-axis. The terminal side is the ray that rotates away from the initial side by the angle's measure.

• Positive angles rotate counterclockwise from the initial side.
• Negative angles rotate clockwise from the initial side.

Quadrantal angles are special cases where the terminal side lands exactly on one of the coordinate axes:

• 0° (or 0 radians): positive x-axis
• 90° (or $\frac{\pi}{2}$ radians): positive y-axis
• 180° (or $\pi$ radians): negative x-axis
• 270° (or $\frac{3\pi}{2}$ radians): negative y-axis

A reference angle is the acute angle (always between 0° and 90°) formed between the terminal side and the x-axis. Reference angles matter because they let you relate trig values of any angle back to a first-quadrant angle. For example, the reference angle for 150° is 30°, since the terminal side is 30° away from the negative x-axis.

Degrees vs. Radians Conversion

Degrees and radians are just two different units for the same thing: the amount of rotation. A full rotation equals 360° or $2\pi$ radians. That relationship gives you both conversion factors.

Degrees → Radians: Multiply by $\frac{\pi}{180}$

Radians → Degrees: Multiply by $\frac{180}{\pi}$

A helpful benchmark: $\pi$ radians = 180°. If you memorize that single fact, you can derive every conversion. For instance, $\frac{\pi}{6}$ is one-sixth of 180°, which is 30°.

Coterminal Angles and Applications

Coterminal angles share the same terminal side, meaning they point in the exact same direction even though their measures differ.

To find coterminal angles:

• In degrees: Add or subtract multiples of 360°.
• In radians: Add or subtract multiples of $2\pi$.

Why does this matter? Coterminal angles produce identical trigonometric function values. So if you're asked to evaluate a trig function at a large or negative angle, you can find a simpler coterminal angle first.

To find the coterminal angle between 0° and 360° (or between 0 and $2\pi$), keep adding or subtracting full rotations until you land in that range.

Arc Length Calculation

An arc is a portion of a circle's circumference, and its length depends on both the central angle and the radius.

Using radians (the cleaner formula):

$s = r\theta$

where $s$ is arc length, $r$ is the radius, and $\theta$ is the central angle in radians.

Using degrees:

$s = \frac{\theta}{360} \times 2\pi r$

The radian formula is simpler, and that's actually why radians exist. The radian was defined so that arc length would just be radius times angle.

Linear and Angular Speed

When an object moves along a circular path, you can describe its motion two ways.

Angular speed ($\omega$) measures how fast the angle changes:

$\omega = \frac{\theta}{t}$

where $\theta$ is the angle of rotation (in radians) and $t$ is time. Units are typically rad/s.

Linear speed ($v$) measures how fast a point on the edge actually travels:

$v = \frac{s}{t}$

where $s$ is the arc length traveled. Units are typically m/s.

These two quantities are connected through the radius:

$v = \omega \times r$

This means that for the same angular speed, a point farther from the center moves faster. Think about a merry-go-round: a horse on the outer edge covers more distance per revolution than one near the center, even though both complete the rotation in the same time.

Special Angle Relationships

• Complementary angles sum to 90° (or $\frac{\pi}{2}$ radians).
• Supplementary angles sum to 180° (or $\pi$ radians).

These relationships show up constantly in trig identities. For instance, the sine of an angle equals the cosine of its complement: $\sin(\theta) = \cos(90° - \theta)$. That's where the name "co-sine" comes from.

Angles of elevation and depression come up in applied problems:

• An angle of elevation is measured upward from a horizontal line to your line of sight.
• An angle of depression is measured downward from a horizontal line to your line of sight.

One useful fact: if you're looking down at an object at an angle of depression of 25°, then from that object's perspective, the angle of elevation up to you is also 25° (they're alternate interior angles formed by a horizontal line and the line of sight).