Trigonometric Functions, Identities, and Transformations

Radian and Degree Conversions

• Conversion formula from radians to degrees: $1\text{ radian} = \left(\frac{180}{\pi}\right)^\circ \approx 57.3^\circ$

• Conversion formula from degrees to radians: $1^\circ = \frac{\pi}{180}\text{ radians} \approx 0.017\text{ radians}$

The Unit Circle and Angle Values

• Angle $0^\circ$ ($0\text{ rad}$): $\left(\cos(0), \sin(0)\right) = (1, 0)$

• Angle $30^\circ$ (\left(\frac{\pi}{6}\text{ rad}\right)): $\left(\cos\left(\frac{\pi}{6}\right), \sin\left(\frac{\pi}{6}\right)\right) = \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

• Angle $45^\circ$ (\left(\frac{\pi}{4}\text{ rad}\right)): $\left(\cos\left(\frac{\pi}{4}\right), \sin\left(\frac{\pi}{4}\right)\right) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

• Angle $60^\circ$ (\left(\frac{\pi}{3}\text{ rad}\right)): $\left(\cos\left(\frac{\pi}{3}\right), \sin\left(\frac{\pi}{3}\right)\right) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$

• Angle $90^\circ$ (\left(\frac{\pi}{2}\text{ rad}\right)): $\left(\cos\left(\frac{\pi}{2}\right), \sin\left(\frac{\pi}{2}\right)\right) = (0, 1)$

• Angle $120^\circ$ (\left(\frac{2\pi}{3}\text{ rad}\right)): $\left(\cos\left(\frac{2\pi}{3}\right), \sin\left(\frac{2\pi}{3}\right)\right) = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$

• Angle $135^\circ$ (\left(\frac{3\pi}{4}\text{ rad}\right)): $\left(\cos\left(\frac{3\pi}{4}\right), \sin\left(\frac{3\pi}{4}\right)\right) = \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

• Angle $150^\circ$ (\left(\frac{5\pi}{6}\text{ rad}\right)): $\left(\cos\left(\frac{5\pi}{6}\right), \sin\left(\frac{5\pi}{6}\right)\right) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

• Angle $180^\circ$ (\left(\pi\text{ rad}\right)): $\left(\cos(\pi), \sin(\pi)\right) = (-1, 0)$

• Angle $210^\circ$ (\left(\frac{7\pi}{6}\text{ rad}\right)): $\left(\cos\left(\frac{7\pi}{6}\right), \sin\left(\frac{7\pi}{6}\right)\right) = \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$

• Angle $225^\circ$ (\left(\frac{5\pi}{4}\text{ rad}\right)): $\left(\cos\left(\frac{5\pi}{4}\right), \sin\left(\frac{5\pi}{4}\right)\right) = \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$

• Angle $240^\circ$ (\left(\frac{4\pi}{3}\text{ rad}\right)): $\left(\cos\left(\frac{4\pi}{3}\right), \sin\left(\frac{4\pi}{3}\right)\right) = \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$

• Angle $270^\circ$ (\left(\frac{3\pi}{2}\text{ rad}\right)): $\left(\cos\left(\frac{3\pi}{2}\right), \sin\left(\frac{3\pi}{2}\right)\right) = (0, -1)$

• Angle $300^\circ$ (\left(\frac{5\pi}{3}\text{ rad}\right)): $\left(\cos\left(\frac{5\pi}{3}\right), \sin\left(\frac{5\pi}{3}\right)\right) = \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$

• Angle $330^\circ$ (\left(\frac{11\pi}{6}\text{ rad}\right)): $\left(\cos\left(\frac{11\pi}{6}\right), \sin\left(\frac{11\pi}{6}\right)\right) = \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$

• Angle $360^\circ$ (\left(2\pi\text{ rad}\right)): $\left(\cos(2\pi), \sin(2\pi)\right) = (1, 0)$

Periodic Functions and Symmetry

• Definition of Periodic Function: A function $f$ is periodic if there exists a positive number $p$ such that: $f(x + p) = f(x)$ for all $x$ in the domain of $f$.

• Fundamental Period: The smallest positive value of $p$ for which $f(x + p) = f(x)$ is called the period of $f$

• Independent Variable Convention: When graphing functions in the coordinate plane, the independent variable is usually denoted by $x$

• Fundamental Periods of Trigonometric Functions:

  • The tangent and cotangent functions have a fundamental period of $\pi$
  • The sine, cosine, secant, and cosecant functions have a fundamental period of $2\pi$

• Parity and Symmetry Properties:

  • Even Functions (Symmetric with respect to the y-axis): $\cos(-x) = \cos(x)$$\sec(-x) = \sec(x)$
  • Odd Functions (Symmetric with respect to the origin): $\sin(-x) = -\sin(x)$$\csc(-x) = -\csc(x)$$\tan(-x) = -\tan(x)$$\cot(-x) = -\cot(x)$

Fundamental Trigonometric Identities

• Primary Pythagorean Identity: $\cos^2(\theta) + \sin^2(\theta) = 1$

• Tangent-Secant Pythagorean Identity: $1 + \tan^2(\theta) = \sec^2(\theta)$

• Cotangent-Cosecant Pythagorean Identity: $1 + \cot^2(\theta) = \csc^2(\theta)$

Transformations of Trigonometric Graphs

• Standard Transformation Form: $y = A\sin(B(x - C)) + D$ or $y = A\cos(B(x - C)) + D$

• Component Descriptions:

  • Vertical Stretch / Amplitude: $|A|$ represents the amplitude of the function.
  • Axis Reflection: A negative value of $A$ results in a vertical reflection across the x-axis.
  • Horizontal Stretch / Compression and Period: The parameter $B$ affects the period $p$, given by: $p = \frac{2\pi}{|B|}$
  • Horizontal Shift (Phase Shift): $C$ represents the horizontal shift along the x-axis.
  • Vertical Shift / Midline: $D$ represents the vertical shift, moving the horizontal centerline axis to $y = D$

Special Trigonometric Inequalities

• For an angle $x$ measured in radians, the sine and cosine functions satisfy the following boundary inequalities: $-|x| \le \sin(x) \le |x|$$1 - \cos(x) \le |x|$$1 - \frac{x^2}{2} \le \cos(x) \le 1$

Double-Angle and Half-Angle Formulas

• Double-Angle Formulas:

  • Sine double-angle formula: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
  • Cosine double-angle formulas: $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$$\cos(2\theta) = 2\cos^2(\theta) - 1$$\cos(2\theta) = 1 - 2\sin^2(\theta)$
  • Tangent double-angle formula: $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

• Half-Angle and Power-Reducing Formulas:

  • Sine half-angle and power-reducing identity: $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$$\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$
  • Cosine half-angle and power-reducing identity: $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$$\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$

Law of Cosines

• Relation for Arbitrary Triangles: If $a$, $b$, and $c$ are the side lengths of a triangle $ABC$, and $\theta$ is the angle opposite side $c$: $c^2 = a^2 + b^2 - 2ab\cos(\theta)$