5: Trigonometric functions review

Trigonometric Functions Review

Trigonometric functions are periodic functions essential for modeling oscillations, waves, and circular motion. They are defined using the unit circle (radius 1) or right triangles:

• Sine (sin θ): $y$-coordinate on the unit circle, or opposite/hypotenuse.
• Cosine (cos θ): $x$-coordinate, or adjacent/hypotenuse.
• Tangent (tan θ): sin θ/cos θ, or opposite/adjacent.
  Reciprocal functions include cosecant (csc θ), secant (sec θ), and cotangent (cot θ).

Key Properties and Graphs

• Periodicity:
  • $\sin(\theta + 2\pi) = \sin \theta$, $\cos(\theta + 2\pi) = \cos \theta$ (period $2\pi$).
  • $\tan(\theta + \pi) = \tan \theta$ (period $\pi$).
• Amplitude and Range:
  • sin and cos: Amplitude $= 1$, Range $[-1, 1]$.
  • tan: No amplitude, Range $(-\infty, \infty)$, with vertical asymptotes at $\theta = \frac{\pi}{2} + k\pi$.
• Graph Characteristics:
  • sin θ: Starts at $(0,0)$, peaks at $(\pi/2, 1)$.
  • cos θ: Starts at $(0,1)$, crosses zero at $\pi/2$.
  • tan θ: Asymptotes at $\pi/2$, crosses $(0,0)$.

Fundamental Identities

Memorize these for integration/differentiation:

• Pythagorean:
  $\sin^2\theta + \cos^2\theta = 1$,
  $1 + \tan^2\theta = \sec^2\theta$,
  $1 + \cot^2\theta = \csc^2\theta$.
• Reciprocal:
  $\csc\theta = \frac{1}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$, $\cot\theta = \frac{1}{\tan\theta}$.
• Even-Odd:
  $\sin(-\theta) = -\sin \theta$ (odd), $\cos(-\theta) = \cos \theta$ (even), $\tan(-\theta) = -\tan \theta$ (odd).
• Co-function:
  $\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta$, $\tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta$.

Special Angles

Recall exact values for common angles using the unit circle or triangles:

θ (rad)	$0$	$\pi/6$	$\pi/4$	$\pi/3$	$\pi/2$
sin θ	$0$	$1/2$	$\sqrt{2}/2$	$\sqrt{3}/2$	$1$
cos θ	$1$	$\sqrt{3}/2$	$\sqrt{2}/2$	$1/2$	$0$
tan θ	$0$	$\sqrt{3}/3$	$1$	$\sqrt{3}$	undef.

Solving Basic Equations

Solve equations like $\sin x = a$ ($-1 \leq a \leq 1$) using reference angles and symmetry:

1. Find solutions in $[0, 2\pi)$:
   • $\sin x = a$ → Principal solution $x = \arcsin a$, second solution $x = \pi - \arcsin a$.
   • $\cos x = a$ → $x = \arccos a$ and $x = 2\pi - \arccos a$.
2. Extend to all reals by adding $2k\pi$ (or $k\pi$ for tan).
   Example: $\sin x = \frac{1}{2}$ → Solutions: $x = \frac{\pi}{6} + 2k\pi$, $x = \frac{5\pi}{6} + 2k\pi$.

Ensure fluency with these concepts, as they underpin derivatives, integrals, and applications in later sections.