Trigonometric Functions

Learning Objectives

• Understand the unit-circle definition of trigonometric functions such as $\sin x$, $\cos x$, and $\tan x$, and their reciprocal functions.
• Calculate and convert angle measurements between degrees and radians, and comprehend their implications for trigonometric functions.
• Analyze the graphical representation of trigonometric functions, focusing on periodicity, amplitude, and phase shifts.
• Determine the domain and range of trigonometric functions, considering restrictions due to asymptotes and periodicity.
• Solve trigonometric equations using identities and algebraic manipulation, including finding general solutions.
• Derive and apply formulas for trigonometric functions of multiple angles, such as double and triple angle formulas.
• Identify the signs of all six trigonometric functions in different quadrants and use this information to deduce other function values.
• Explore the periodicity of trigonometric functions and reduce large or negative angles using multiples of $2\pi$, $\pi$, or $360^\circ$.
• Apply sum and difference identities for $\sin(x \pm y)$, $\cos(x \pm y)$, $\tan(x \pm y)$, and $\cot(x \pm y)$ in evaluations and proofs.
• Utilize product-to-sum and sum-to-product identities in proofs and simplifications of trigonometric expressions.
• Prove and simplify expressions using fundamental, compound-angle, multiple-angle, and sum-product identities.

Chapter Notes

Trigonometric Functions and Identities

• Unit-circle definition: For a point $P(a, b)$ on the unit circle, $\cos x = a$ and $\sin x = b$.
• Fundamental identities:
  • $\sin^2 x + \cos^2 x = 1$
  • $1 + \tan^2 x = \sec^2 x$
  • $1 + \cot^2 x = \csc^2 x$
• Quadrantal angles:
  • $\cos 0^\circ = 1$, $\sin 0^\circ = 0$
  • $\cos \frac{\pi}{2} = 0$, $\sin \frac{\pi}{2} = 1$
  • $\cos \pi = -1$, $\sin \pi = 0$
  • $\cos \frac{3\pi}{2} = 0$, $\sin \frac{3\pi}{2} = -1$
  • $\cos 2\pi = 1$, $\sin 2\pi = 0$

Angle Measurement and Conversion

• Degree measure: One complete revolution is $360^\circ$.
• Radian measure: One complete revolution is $2\pi$ radians.
• Conversion:
  • $1^\circ = \frac{\pi}{180}$ radians
  • $1$ radian $= \frac{180}{\pi}^\circ$

Graphical Representation of Trigonometric Functions

• Sine and Cosine:
  • Amplitude: 1
  • Period: $2\pi$
• Tangent and Cotangent:
  • Period: $\pi$
• Secant and Cosecant:
  • Period: $2\pi$

Domain and Range of Trigonometric Functions

• Sine and Cosine:
  • Domain: All real numbers
  • Range: $[-1, 1]$
• Tangent and Cotangent:
  • Domain: $x \neq (2n+1)\frac{\pi}{2}$ for tangent, $x \neq n\pi$ for cotangent
  • Range: All real numbers
• Secant and Cosecant:
  • Domain: $x \neq (2n+1)\frac{\pi}{2}$ for secant, $x \neq n\pi$ for cosecant
  • Range: $(-\infty, -1] \cup [1, \infty)$

Trigonometric Equations and Solutions

• General solutions:
  • $\sin x = 0 \Rightarrow x = n\pi$
  • $\cos x = 0 \Rightarrow x = (2n+1)\frac{\pi}{2}$

Trigonometric Functions of Multiple Angles

• Double angle formulas:
  • $\sin 2x = 2\sin x \cos x$
  • $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$
  • $\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$

Signs and Quadrants of Trigonometric Functions

• Quadrant I: All positive
• Quadrant II: Sine positive
• Quadrant III: Tangent positive
• Quadrant IV: Cosine positive

Periodicity and Reduction of Angles

• Periodicity:
  • $\sin(x + 2n\pi) = \sin x$
  • $\cos(x + 2n\pi) = \cos x$
  • $\tan(x + n\pi) = \tan x$

Sum and Difference Identities

• Sine and Cosine:
  • $\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$
  • $\cos(x \pm y) = \cos x \cos y \mp \sin x \sin y$

Product-to-Sum and Sum-to-Product Identities

• Identities:
  • $\cos x + \cos y = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$
  • $\sin x + \sin y = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$

Trigonometric Identity Proofs and Simplification

• Example:
  • Prove $\cos(x + y) = \cos x \cos y - \sin x \sin y$ using the unit circle and congruent triangles.