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).

This quiz focuses on the properties of the sine and cosine functions, essential elements of circular functions in trigonometry. Explore definitions, ranges, periodicity, symmetry, key values, and graph characteristics to deepen your understanding of these fundamental concepts.