Essential Concepts

Angles and Angle Measurement

• An angle is the union of two rays having a common endpoint
• Greek letters ([latex]\theta, \phi, \alpha, \beta, \gamma[/latex]) are commonly used to represent angle measures
• The measure of an angle is the amount of rotation from the initial side to the terminal side
  • Counterclockwise rotation produces a positive angle
  • Clockwise rotation produces a negative angle
• An angle is in standard position when its vertex is at the origin and its initial side extends along the positive x-axis
• The coordinate plane is divided into four quadrants, numbered I through IV counterclockwise starting from the positive x-axis

Degrees and Radians

• A complete circular rotation contains 360°
• Always include the degree symbol (°) or the word “degrees”
• One radian is the measure of a central angle that intercepts an arc equal in length to the radius
• A full revolution equals [latex]2\pi[/latex] radians
• When an angle is described without units, it refers to radian measure
• Use the proportion: [latex]\frac{\theta}{180} = \frac{\theta^R}{\pi}[/latex]
  • To convert radians to degrees: multiply by [latex]\frac{180}{\pi}[/latex]
  • To convert degrees to radians: multiply by [latex]\frac{\pi}{180}[/latex]

DMS Form (Degrees, Minutes, Seconds)

• One degree equals 60 minutes: [latex]1° = 60'[/latex]
• One minute equals 60 seconds: [latex]1' = 60"[/latex]
• Written as [latex]D° M' S"[/latex]

Coterminal Angles

• Two angles in standard position with the same terminal side
  • To find coterminal angles in degrees: add or subtract multiples of 360°
  • To find coterminal angles in radians: add or subtract multiples of [latex]2\pi[/latex]

Reference Angles

• The measure of the smallest positive acute angle formed by the terminal side and the horizontal axis
  • Always between 0° and 90° (or 0 and [latex]\frac{\pi}{2}[/latex] radians)
  • For Quadrant II or III: [latex]|\pi - t|[/latex] or [latex]|180° - t|[/latex]
  • For Quadrant IV: [latex]2\pi - t[/latex] or [latex]360° - t[/latex]

Complementary and Supplementary Angles

• Complementary angles sum to 90° (or [latex]\frac{\pi}{2}[/latex] radians)
• Supplementary angles sum to 180° (or [latex]\pi[/latex] radians)
• An angle can only have a complement if it measures less than 90°

Arcs and Sectors

• In a circle of radius [latex]r[/latex], the length of an arc [latex]s[/latex] subtended by angle [latex]\theta[/latex] (in radians) is: [latex]s = r\theta[/latex]. The angle must be in radians for this formula to work
• The area of a sector with radius [latex]r[/latex] and central angle [latex]\theta[/latex] (in radians) is: [latex]A = \frac{1}{2}\theta r^2[/latex]

Linear and Angular Speed

• Linear speed [latex]v[/latex] is the distance traveled per unit time: [latex]v = \frac{s}{t}[/latex]
• Angular speed [latex]\omega[/latex] is the angular rotation per unit time: [latex]\omega = \frac{\theta}{t}[/latex]

Sine and Cosine Functions

The Unit Circle

• A circle centered at the origin with radius 1
• Any point on the unit circle can be written as [latex](x, y) = (\cos t, \sin t)[/latex]

Sine and Cosine Functions

• For a point [latex](x, y)[/latex] on the unit circle corresponding to angle [latex]t[/latex]:
  • [latex]\cos t = x[/latex] (the x-coordinate)
  • [latex]\sin t = y[/latex] (the y-coordinate)
• Output values are always between -1 and 1

The Pythagorean Identity

• [latex]\cos^2 t + \sin^2 t = 1[/latex]
• Derived from the unit circle equation [latex]x^2 + y^2 = 1[/latex]

Special Angles and Their Values

The Other Trigonometric Functions

• Tangent: [latex]\tan t = \frac{y}{x} = \frac{\sin t}{\cos t}[/latex] (where [latex]x \neq 0[/latex])
• Secant: [latex]\sec t = \frac{1}{x} = \frac{1}{\cos t}[/latex] (where [latex]x \neq 0[/latex])
• Cosecant: [latex]\csc t = \frac{1}{y} = \frac{1}{\sin t}[/latex] (where [latex]y \neq 0[/latex])
• Cotangent: [latex]\cot t = \frac{x}{y} = \frac{\cos t}{\sin t}[/latex] (where [latex]y \neq 0[/latex])

Even and Odd Functions

• Even functions (symmetric about y-axis): [latex]\cos(-t) = \cos t[/latex] and [latex]\sec(-t) = \sec t[/latex]
• Odd functions (symmetric about origin): [latex]\sin(-t) = -\sin t[/latex], [latex]\tan(-t) = -\tan t[/latex], [latex]\csc(-t) = -\csc t[/latex], [latex]\cot(-t) = -\cot t[/latex]

Alternate Pythagorean Identities

• [latex]1 + \tan^2 t = \sec^2 t[/latex]
• [latex]\cot^2 t + 1 = \csc^2 t[/latex]

Period of Trigonometric Functions

• A period is the shortest interval over which a function completes one full cycle
• Sine, cosine, secant, and cosecant have period [latex]2\pi[/latex]
• Tangent and cotangent have period [latex]\pi[/latex]

Key Equations

Arc length	[latex]s = r\theta[/latex]
Area of a sector	[latex]A = \frac{1}{2}\theta r^2[/latex]
Angular speed	[latex]\omega = \frac{\theta}{t}[/latex]
Linear speed	[latex]v = \frac{s}{t}[/latex] or [latex]v = r\omega[/latex]
Pythagorean Identity	[latex]\cos^2 t + \sin^2 t = 1[/latex] [latex]1 + \tan^2 t = \sec^2 t[/latex] [latex]\cot^2 t + 1 = \csc^2 t[/latex]
Tangent function	[latex]\tan t = \frac{\sin t}{\cos t}[/latex]
Secant function	[latex]\sec t = \frac{1}{\cos t}[/latex]
Cosecant function	[latex]\csc t = \frac{1}{\sin t}[/latex]
Cotangent function	[latex]\cot t = \frac{\cos t}{\sin t}[/latex]

Glossary

angle

The union of two rays having a common endpoint called the vertex.

angular speed

The angular rotation [latex]\theta[/latex] per unit time [latex]t[/latex], denoted [latex]\omega = \frac{\theta}{t}[/latex].

arc length

The length of the arc [latex]s[/latex] along a circle subtended by a central angle, calculated as [latex]s = r\theta[/latex] where [latex]\theta[/latex] is in radians.

complementary angles

Two angles whose measures add up to 90° (or [latex]\frac{\pi}{2}[/latex] radians).

cosecant

The reciprocal of the sine function: [latex]\csc t = \frac{1}{\sin t}[/latex].

cosine

For a point [latex](x, y)[/latex] on the unit circle corresponding to angle [latex]t[/latex], [latex]\cos t = x[/latex].

cotangent

The reciprocal of the tangent function: [latex]\cot t = \frac{\cos t}{\sin t}[/latex].

coterminal angles

Two angles in standard position that have the same terminal side.

degree

A unit of angle measurement where one degree is [latex]\frac{1}{360}[/latex] of a circular rotation.

DMS form

Notation for expressing angles using degrees, minutes, and seconds (D° M’ S”), where 1° = 60′ and 1′ = 60″.

even function

A function where [latex]f(-x) = f(x)[/latex]. Cosine and secant are even trigonometric functions.

initial side

The fixed ray from which angle measurement begins, extending along the positive x-axis in standard position.

linear speed

The distance traveled per unit time, calculated as [latex]v = \frac{s}{t}[/latex] or [latex]v = r\omega[/latex].

odd function

A function where [latex]f(-x) = -f(x)[/latex]. Sine, tangent, cosecant, and cotangent are odd trigonometric functions.

period

The shortest interval [latex]P[/latex] over which a function completes one full cycle, where [latex]f(x + P) = f(x)[/latex].

quadrantal angles

Angles whose terminal side lies on an axis, including 0°, 90°, 180°, 270°, and 360°.

radian

The measure of a central angle that intercepts an arc equal in length to the radius of the circle. A full revolution equals [latex]2\pi[/latex] radians.

reference angle

The measure of the smallest positive acute angle formed by the terminal side of an angle and the horizontal axis.

secant

The reciprocal of the cosine function: [latex]\sec t = \frac{1}{\cos t}[/latex].

sector

A region of a circle bounded by two radii and the intercepted arc, with area [latex]A = \frac{1}{2}\theta r^2[/latex].

sine

For a point [latex](x, y)[/latex] on the unit circle corresponding to angle [latex]t[/latex], [latex]\sin t = y[/latex].

standard position

An angle positioned with its vertex at the origin and its initial side extending along the positive x-axis.

supplementary angles

Two angles whose measures add up to 180° (or [latex]\pi[/latex] radians).

tangent

The ratio of sine to cosine: [latex]\tan t = \frac{\sin t}{\cos t}[/latex].

terminal side

The rotated ray of an angle after rotation from the initial side.

unit circle

A circle centered at the origin with radius 1, where [latex]x^2 + y^2 = 1[/latex].