Trigonometry Review + Derivatives of Trigonometric Functions

The Unit Circle

The Unit Circle is defined mathematically as the set of points that satisfy the equation:
x^2 + y^2 = 1
This circle is centered at the origin (0,0) with a radius of 1.
The unit circle is essential for understanding trigonometric functions because it relates angles, arc lengths, and coordinates.

The unit circle allows for the definition of the sine and cosine functions as follows:
- For an angle \theta, the coordinates of the point P(\theta) on the unit circle are:
- \cos(\theta) (the x-coordinate)
- \sin(\theta) (the y-coordinate)
Radian Measure: One radian is the angle that subtends an arc length of 1 on the unit circle. The full angle in radians for a circle is given by:
- 2\pi radians (which equals 360 degrees).
Positive angles on the unit circle are measured in the counterclockwise direction, while negative angles are measured clockwise.

Sinusoidal Functions are defined as functions whose graphs exhibit a wave-like pattern resembling sine or cosine curves.
The general form of a sinusoidal function is given by:
- f(x) = A\sin(Bx) + D
- or
- g(x) = A\cos(Bx) + D
- Where:
  - A = amplitude (half the distance between maximum and minimum values)
  - B = affects the period of the function
  - D = vertical shift (average value)
More generally, sinusoidal functions can also be expressed in terms of phase shifts:
- f(x) = A\sin(B(x + C)) + D
- g(x) = A\cos(B(x + C)) + D
- Where C represents the phase shift.
To graph a phase-shifted sinusoidal function, one should first account for the period (stretch or compress horizontally) and then apply the phase shift (horizontal shift).

An application in modeling can be given by the heart volume example:
- The average volume of a heart is 140 milliliters (ml), and it ejects approximately half of its volume with each beat (70 ml).
- For a trained athlete with a heartbeat frequency of 50 bpm, you can model the volume of blood in the heart (as a function of time t) using a sinusoidal function.
- If t measures time in minutes, with t = 0 as the moment when the heart is at maximum volume, the volume can be modeled as:
- V(t) = A sin(Bt) + D

The Tangent Function:
- Defined as the ratio of sine over cosine:
- tan(\theta) = \frac{sin(\theta)}{cos(\theta)}
Three additional basic trigonometric functions are defined as multiplicative inverses:
- Secant: sec(\theta) = \frac{1}{cos(\theta)}
- Cosecant: csc(\theta) = \frac{1}{sin(\theta)}
- Cotangent: cot(\theta) = \frac{1}{tan(\theta)}

Consider an angle \theta in a right triangle (where one angle is exactly 90 degrees or \frac{\pi}{2} radians).
- The angle \theta must be acute, such that:
- 0 < \theta < \frac{\pi}{2} or equivalently, 0 < \theta < 90 degrees.
Fundamental definitions of trigonometric functions in a right triangle include:
- sin(\theta) = \frac{opposite}{hypotenuse}
- cos(\theta) = \frac{adjacent}{hypotenuse}
- tan(\theta) = \frac{opposite}{adjacent}
The Pythagorean Identity forms the cornerstone of right triangle trigonometry:
- sin^2(\theta) + cos^2(\theta) = 1

Familiarity with all six basic trigonometric functions and their domains and ranges, most importantly how they relate to sin and cos.
Understanding the periods and representations of graphs for sin(x), cos(x), and tan(x).
Knowledge of trigonometric function values at common angles (ex. sin(\frac{\pi}{6}), sin(\frac{\pi}{4}), sin(\frac{\pi}{3})).
Skill in graphing sinusoidal functions and modeling given scenarios into sinusoidal functions.
Mastery of definitions of trigonometric functions via right triangles.
Application of the Pythagorean identity.
Understanding of similar triangles.
Ability to compute the area of a triangle (as well as other shapes like rectangles and circles) and their respective volumes.

Importance of Radians: For angle \theta measured in radians, the following limit theorem holds:
- \lim_{\theta \to 0} \frac{sin(\theta)}{\theta} = 1
- This signifies that for small angles, sin(\theta) is approximately equal to \theta.

Derivative of sin(x):
- Required identities:
  - Identity: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
  - Important limits:
    - \lim_{h \to 0} \frac{sin(h)}{h} = 1
    - \lim_{h \to 0} \cos(h) = 1
- Result:
- \frac{d}{dx}(sin(x)) = cos(x)
Derivative of cos(x):
- Notably, cos(x) can be expressed as a phase shift of sin(x):
- cos(x) = sin(\left(x + \frac{\pi}{2}\right)).
- Result:
- \frac{d}{dx}(cos(x)) = -sin(x)
Derivative of tan(x):
- Utilizing the derivatives of sin and cos, we find:
- \frac{d}{dx}(tan(x)) = sec^2(x)
Complete Derivative Theorems:
- The established derivatives include:
  - \frac{d}{dx}(sin(x)) = cos(x)
  - \frac{d}{dx}(cos(x)) = -sin(x)
  - \frac{d}{dx}(tan(x)) = sec^2(x)
  - \frac{d}{dx}(cot(x)) = -csc^2(x)
  - \frac{d}{dx}(sec(x)) = sec(x)tan(x)
  - \frac{d}{dx}(csc(x)) = -csc(x)cot(x)

Example:
- If a particle's position on the x-axis is given by the function s(t) = sin(t^4) where s is in meters and t in seconds, evaluate whether the particle is speeding up, slowing down, or moving at a constant speed at t = 1 second.