Trigonometry: Sine and Cosine Functions

Circular Functions

Sine Function Properties

• Definition: Sine function (sin) relates to the ratio of the opposite side to the hypotenuse in a right triangle.
• Range: ([-1, 1])
• Periodicity: Period of (2\pi) radians (360 degrees).
• Symmetry: Odd function; ( \sin(-x) = -\sin(x) )
• Key Values:
  • (\sin(0) = 0)
  • (\sin\left(\frac{\pi}{2}\right) = 1)
  • (\sin(\pi) = 0)
  • (\sin\left(\frac{3\pi}{2}\right) = -1)
• Graph Characteristics:
  • Continuous wave-like pattern.
  • Intercepts at integer multiples of (\pi).

Cosine Function Properties

• Definition: Cosine function (cos) relates to the ratio of the adjacent side to the hypotenuse in a right triangle.
• Range: ([-1, 1])
• Periodicity: Period of (2\pi) radians (360 degrees).
• Symmetry: Even function; ( \cos(-x) = \cos(x) )
• Key Values:
  • (\cos(0) = 1)
  • (\cos\left(\frac{\pi}{2}\right) = 0)
  • (\cos(\pi) = -1)
  • (\cos\left(\frac{3\pi}{2}\right) = 0)
• Graph Characteristics:
  • Similar wave pattern, shifted horizontally compared to sine.
  • Intercepts at odd multiples of (\frac{\pi}{2}).

Tangent Function Properties

• Definition: Tangent function (tan) is the ratio of sine to cosine: ( \tan(x) = \frac{\sin(x)}{\cos(x)} ).
• Range: ((-\infty, \infty))
• Periodicity: Period of (\pi) radians (180 degrees).
• Asymptotes: Vertical asymptotes where (\cos(x) = 0) (e.g., (x = \frac{\pi}{2} + n\pi), (n \in \mathbb{Z})).
• Symmetry: Odd function; ( \tan(-x) = -\tan(x) )
• Key Values:
  • (\tan(0) = 0)
  • (\tan\left(\frac{\pi}{4}\right) = 1)
  • (\tan\left(\frac{3\pi}{4}\right) = -1)
• Graph Characteristics:
  • Repeats every (\pi) radians.
  • Displays increasing behaviour across each period.

Inverse Circular Functions

• Definition: Provide angles corresponding to a given value of a circular function.
• Notations:
  • Arcsine: (y = \sin^{-1}(x)) or (y = \text{arcsin}(x))
  • Arccosine: (y = \cos^{-1}(x)) or (y = \text{arccos}(x))
  • Arctangent: (y = \tan^{-1}(x)) or (y = \text{arctan}(x))
• Ranges:
  • (y = \text{arcsin}(x): [-\frac{\pi}{2}, \frac{\pi}{2}])
  • (y = \text{arccos}(x): [0, \pi])
  • (y = \text{arctan}(x): \left(-\frac{\pi}{2}, \frac{\pi}{2}\right))
• Key Points:
  • Inverse functions reverse the effect of the original function.
  • Useful in solving equations involving circular functions.
  • Visualized through reflection over the line (y = x) in graphs.

Circular Functions

• The sine, cosine, and tangent functions are fundamental trigonometric functions.
• Each function relates to specific ratios of sides in a right triangle.
• Sine (sin) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of sine to cosine, or equivalently, the opposite side to the adjacent side.
• All three functions have specific ranges, periods, and symmetries.

Sine Function Properties

• The sine function has a range of -1 to 1.
• It has a period of (2\pi) radians or 360 degrees.
• This means the function's graph repeats every (2\pi) radians.
• The sine function is an odd function, meaning ( \sin(-x) = -\sin(x) ).
• The sine function is symmetrical about the origin.
• Key values for the sine function are:
  • (\sin(0) = 0)
  • (\sin\left(\frac{\pi}{2}\right) = 1)
  • (\sin(\pi) = 0)
  • (\sin\left(\frac{3\pi}{2}\right) = -1)

Cosine Function Properties

• The cosine function also has a range of -1 to 1.
• Similar to sine, it has a period of (2\pi) radians or 360 degrees.
• The cosine function is an even function, meaning ( \cos(-x) = \cos(x) ).
• The cosine function is symmetrical about the y-axis.
• Key values for the cosine function:
  • (\cos(0) = 1)
  • (\cos\left(\frac{\pi}{2}\right) = 0)
  • (\cos(\pi) = -1)
  • (\cos\left(\frac{3\pi}{2}\right) = 0)

Tangent Function Properties

• The tangent function's range is from negative infinity to positive infinity.
• It has a period of (\pi) radians or 180 degrees.
• The tangent function has vertical asymptotes where (\cos(x) = 0), which occurs at (x = \frac{\pi}{2} + n\pi), where (n) is an integer.
• The tangent function is an odd function, meaning ( \tan(-x) = -\tan(x) ).
• It's symmetrical about the origin.
• Key values for the tangent function are:
  • (\tan(0) = 0)
  • (\tan\left(\frac{\pi}{4}\right) = 1)
  • (\tan\left(\frac{3\pi}{4}\right) = -1)

Inverse Circular Functions

• The inverse circular functions, arcsine, arccosine, and arctangent, provide the angle corresponding to a given value of a trigonometric function.
• Inverse functions reverse the effect of the original function.
• Arcsine uses the notation (y = \sin^{-1}(x)) or (y = \text{arcsin}(x)).
• Arccosine is denoted as (y = \cos^{-1}(x)) or (y = \text{arccos}(x))
• Arctangent uses (y = \tan^{-1}(x)) or (y = \text{arctan}(x)).
• The range for the inverse functions is:
  • (y = \text{arcsin}(x): [-\frac{\pi}{2}, \frac{\pi}{2}])
  • (y = \text{arccos}(x): [0, \pi])
  • (y = \text{arctan}(x): \left(-\frac{\pi}{2}, \frac{\pi}{2}\right))
• Inverse circular function are crucial for solving equations involving trigonometric functions.
• Their graphs are represented by reflections of the original functions over the line (y = x).