Trigonometric Functions and Their Properties

Trigonometric Functions

Standard Forms

$$\begin{cases} \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \\\\ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \\\\ \tan \theta = \frac{\sin \theta}{\cos \theta} \end{cases}$$

Basic Properties

Sine Function $\sin(x)$

$$y = \sin x$$

• Period: $2\pi$
• Symmetry: Odd function
• Extremum: $\left( \frac{\pi}{2}+2k\pi, 1 \right)$
• Zeros: $k\pi \ (k \in \mathbb{Z})$

Cosine Function $\cos(x)$

$$y = \cos x$$

• Period: $2\pi$
• Symmetry: Even function
• Extremum: $(2k\pi, 1)$
• Zeros: $\frac{\pi}{2}+k\pi$

Tangent Function $\tan(x)$

$$y = \tan x = \frac{\sin x}{\cos x}$$

• Period: $\pi$
• Asymptotes: $x = \frac{\pi}{2}+k\pi$
• Range: $\mathbb{R}$
• Special points: Passes through origin, symmetric about the center of each period

Core Properties

Derivatives and Integrals

Function	Derivative	Integral
$\sin x$	$\cos x$	$-\cos x + C$
$\cos x$	$-\sin x$	$\sin x + C$
$\tan x$	$\sec^2 x$	$-\ln

Important Identities

Pythagorean Identity:

$$\sin^2x + \cos^2x = 1$$

Sum and Difference Formulas:

$$\sin(a\pm b) = \sin a \cos b \pm \cos a \sin b$$ $$\cos(a \pm b) = \cos{a} \cos{b} \mp \sin{a} \sin{b}$$

Comparison of Functions

Property	Sine Function	Cosine Function	Tangent Function
Initial value	0	1	0
Extremum	Odd multiples of $\pi/2$	Even multiples of $\pi$	None
Periodicity	$2\pi$	$2\pi$	$\pi$
Asymptotes	None	None	Present

	$30^\circ$	$45^\circ$	$60^\circ$
$sin$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$
$cos$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$
$tan$	$\frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$

Reduction Formulas

Summary

1. Split the angle: Express the target angle as $\frac{k\pi}{2} \pm \alpha$
2. Determine parity: Check if $k$ is odd or even to decide whether to change the function name
3. Determine quadrant: Identify the quadrant of $\frac{k\pi}{2} \pm \alpha$ and determine the sign of the new function
4. Combine results: Combine the sign and function name to obtain the final expression

Core Principle Analysis

Odd changes, even remains: Transformation of function names

When the angle takes the form $\frac{k\pi}{2} \pm \alpha$ ($k \in \mathbb{Z}$):

• Odd changes: If $k$ is odd, the trigonometric function name changes to its cofunction (sine↔cosine, tangent↔cotangent)
• Even remains: If $k$ is even, the function name remains unchanged

Example analysis:

1. $\sin\left(\frac{\pi}{2} + \alpha\right)$: $k=1$ (odd), $\sin$ changes to $\cos$ → $\cos\alpha$
2. $\cos(\pi + \alpha)$: $k=2$ (even), name unchanged → $\cos\alpha$
3. $\tan\left(\frac{3\pi}{2} - \alpha\right)$: $k=3$ (odd), $\tan$ changes to $\cot$ → $\cot\alpha$

Sign depends on quadrant: Determining the sign of the function value

Assuming $\alpha$ is an acute angle, treat $\frac{k\pi}{2} \pm \alpha$ as a whole angle, and determine the sign of the result based on its quadrant:

Quadrant sign mnemonic:

All positive in first quadrant, sine in second, tangent and cotangent in third, cosine in fourth

Steps:

1. Determine the quadrant of $\frac{k\pi}{2} \pm \alpha$
2. Determine the sign of the new function based on the quadrant
3. Assign the sign to the simplified result

Example analysis:

• $\cos\left(\pi + \alpha\right)$:

  1. Even remains
  2. $\pi + \alpha$ is in the third quadrant
  3. Cosine is negative in the third quadrant
  4. Result is $-\cos\alpha$

• $\sin\left(\frac{3\pi}{2} - \alpha\right)$:

  1. Odd changes
  2. $\frac{3\pi}{2} - \alpha$ is in the third quadrant
  3. Sine is negative in the third quadrant
  4. Result is $-\cos\alpha$

Comprehensive Application Examples

Example 1: Simplify $\sin\left(\frac{5\pi}{2} - \alpha\right)$

1. Split angle: $\frac{5\pi}{2} = 2\pi + \frac{\pi}{2}$ → $k=5$ (odd)
2. Function name changes: $\sin$ → $\cos$
3. Quadrant determination: $\frac{5\pi}{2} - \alpha$ is in the first quadrant, cosine is positive
4. Result: $\cos\alpha$

Derivation:

$$\sin\left(\frac{5\pi}{2} - \alpha\right) = \sin\left(2\pi + \frac{\pi}{2} - \alpha\right) = \cos\alpha$$

Example 2: Compute $\tan\left(\frac{7\pi}{2} + \alpha\right)$

1. Split angle: $\frac{7\pi}{2} = 3\pi + \frac{\pi}{2}$ → $k=7$ (odd)
2. Function name changes: $\tan$ → $\cot$
3. Quadrant determination: $\frac{7\pi}{2} + \alpha$ is in the fourth quadrant, cotangent is negative
4. Result: $-\cot\alpha$

Derivation:

$$\tan\left(\frac{7\pi}{2} + \alpha\right) = \tan\left(3\pi + \frac{\pi}{2} + \alpha\right) = -\cot\alpha$$

Special Cases

Angles beyond $2\pi$:

• Reduce the angle modulo $2\pi$ before determining. For example, $\frac{9\pi}{2} - \alpha$ simplifies to $\frac{\pi}{2} - \alpha + 4\pi$, so $k=9$ (odd)

Negative angles:

• Use parity to convert. For example, $\sin(-\alpha) = -\sin\alpha$ (odd function property)

Inverse Trigonometric Functions

Standard Forms

$$\begin{cases} \theta = \arcsin\left(\dfrac{\text{opposite}}{\text{hypotenuse}}\right) \\ \theta = \arccos\left(\dfrac{\text{adjacent}}{\text{hypotenuse}}\right) \\ \theta = \arctan\left(\dfrac{\text{opposite}}{\text{adjacent}}\right) \end{cases}$$

Function	Domain	Range	Monotonicity
$\arcsin x$	$[-1,1]$	$[\frac{-\pi}{2},\frac{\pi}{2}]$	Strictly increasing
$\arccos x$	$[-1,1]$	$[0,\pi]$	Strictly decreasing
$\arctan x$	$\mathbb{R}$	$(\frac{-\pi}{2},\frac{\pi}{2})$	Strictly increasing

Basic Properties

Arcsine Function $\arcsin(x)$

$$y = \arcsin x$$

• Domain: $x \in [-1, 1]$
• Range: $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$
• Derivative:

$$\frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}} \quad (-1 < x < 1)$$

• Monotonicity: Strictly increasing
• Special values:
  • $\arcsin(0) = 0$
  • $\arcsin\left( \frac{1}{2} \right) = \frac{\pi}{6}$

Arccosine Function $\arccos(x)$

$$y = \arccos x$$

• Domain: $x \in [-1, 1]$
• Range: $[0, \pi]$
• Derivative:

$$\frac{dy}{dx} = -\frac{1}{\sqrt{1 - x^2}} \quad (-1 < x < 1)$$

• Monotonicity: Strictly decreasing
• Identity:

$$\arccos x + \arcsin x = \frac{\pi}{2}$$

Arctangent Function $\arctan(x)$

$$y = \arctan x$$

• Domain: $\mathbb{R}$
• Range: $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$
• Derivative:

$$\frac{dy}{dx} = \frac{1}{1 + x^2}$$

• Asymptotes: $y = \pm \frac{\pi}{2}$
• Symmetry:

$$\arctan(-x) = -\arctan x$$

Typical applications:
As $x \to \infty$, $\arctan x \to \frac{\pi}{2}$
Integration formula:

$$\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \arctan\left( \frac{x}{a} \right) + C \quad (a > 0)$$

Effects of Parameters on Graphs

Taking $\sin(x)$ as an example:

Basic Parameters

Amplitude Modulation:

$$y = A\sin x \quad \Rightarrow \quad \text{Peak height} |A|$$

Frequency Change:

$$y = \sin \omega x \quad \Rightarrow \quad \text{Period} T = \frac{2\pi}{\omega}$$

Phase Shift:

$$y = \sin(x + \phi) \quad \Rightarrow \quad \text{Graph shifts left by }\phi$$

Composite Waveform Example:

$$y = 2\sin(3x - \pi/4) \quad \text{Properties: amplitude 2, frequency 3, right shift }\frac{\pi}{12}$$

Graph Transformations

Horizontal Translation (Phase Shift)

Mathematical expression:

$$y = \sin(x + c)$$

• $c < 0$: Graph shifts right by $c$ units
• $c > 0$: Graph shifts left by $|c|$ units
  Example:

$$y = \sin\left(x - \frac{\pi}{2}\right) \quad \Rightarrow \quad \text{Original peak } \left(\frac{\pi}{2},1\right) \text{ becomes } (\pi,1)$$

Vertical Translation

Mathematical expression:

$$y = \sin x + d$$

• $d > 0$: Graph shifts up by $d$ units
• $d < 0$: Graph shifts down by $|d|$ units
  Example:

$$y = \sin x + 2 \quad \Rightarrow \quad \text{Peak value } 3 \text{, trough value } 1$$

Amplitude Modulation

Mathematical expression:

$$y = A \sin x$$

• $|A| > 1$: Vertical stretch
• $0 < |A| < 1$: Vertical compression
• $A < 0$: Reflection over the $x$-axis
  Example:

$$y = -3 \sin x \quad \Rightarrow \quad \text{Amplitude } 3 \text{, waveform inverted}$$

$$y = \frac{1}{2} \sin x \quad \Rightarrow \quad \text{Amplitude } \frac{1}{2} \text{, waveform vertically compressed}$$

Period Adjustment (Frequency Change)

Mathematical expression:

$$y = \sin(Bx)$$

• Period: $T = \frac{2\pi}{|B|}$
• $|B| > 1$: Horizontal compression (shorter period)
• $0 < |B| < 1$: Horizontal stretch (longer period)
  Example:

$$y = \sin(2x) \quad \Rightarrow \quad T = \pi \text{ (half of original period } 2\pi\text{)}$$

$$y = \sin(\frac{1}{2}x) \quad \Rightarrow \quad T = 4\pi \text{ (twice the original period } 2\pi\text{)}$$

Reflection Transformations

1. Reflection over the $x$-axis
   Expression:

   $$y = -\sin x \quad \Rightarrow \quad \text{All y-coordinates are negated}$$

   
   Example:

   $$\sin\left(\frac{\pi}{2}\right) = 1 \quad \Rightarrow \quad -\sin\left(\frac{\pi}{2}\right) = -1$$

2. Reflection over the $y$-axis
   Expression:

   $$y = \sin(-x) \quad \Rightarrow \quad \text{Equivalent to } -\sin x \ (\because \sin(-x) = -\sin x)$$

   
   Example:

   $$\sin\left(-\frac{\pi}{2}\right) = -1$$

Composite Transformation Example

Example Function:

$$y = 3 \sin\left(2x - \frac{\pi}{4}\right) + 1$$

1. Amplitude: $3$
2. Period: $T = \frac{2\pi}{2} = \pi$
3. Phase shift: $\frac{\pi/4}{2} = \frac{\pi}{8}$ (right shift by $\frac{\pi}{8}$)
4. Vertical translation: Upward by $1$ unit

Final effect:

• Starting point changes from $(0,0)$ to $\left(\frac{\pi}{8},1\right)$
• Maximum point: $\left(\frac{\pi}{8} + \frac{\pi}{4},4\right)$
• Minimum point: $\left(\frac{\pi}{8} + \frac{3\pi}{4},-2\right)$