Graphs of Trigonometric Functions

Graphing Sine and Cosine Functions

Understanding the graphs of sine and cosine functions is fundamental in trigonometry. These periodic functions are characterized by their amplitude, period, phase shift, and vertical shift.

• Amplitude: The maximum distance from the midline to the peak (for y = a sin(x), amplitude is |a|).

• Period: The length of one complete cycle. For y = sin(bx), period is .

• Phase Shift: Horizontal shift left or right.

• Vertical Shift: Up or down movement of the graph.

Example: For , the amplitude is 2, period is , and there is no phase or vertical shift.

Example: For , the amplitude is 2, period is , and the graph is reflected over the x-axis.

Evaluating Trigonometric Functions

Exact Values of Sine, Cosine, and Tangent

Trigonometric functions have well-known values at special angles (0, , , , , etc.). These values are essential for solving equations and simplifying expressions.

• sin(0) = 0

• sin() =

• sin() =

• sin() =

• sin() = 1

Similar values exist for cosine and tangent. These can be organized in a table for quick reference.

Trigonometric Identities

Fundamental Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined. They are used to simplify expressions and solve equations.

• Pythagorean Identity:

• Quotient Identities: ,

• Reciprocal Identities: , ,

Sum and Difference Formulas

These formulas allow the calculation of trigonometric functions of sums or differences of angles.

Example:

Double Angle and Half Angle Formulas

These identities are useful for expressing trigonometric functions of double or half angles in terms of single angles.

Solving Trigonometric Equations

General Strategies

To solve trigonometric equations, use identities to simplify the equation, isolate the trigonometric function, and solve for the variable within the given interval.

• Use algebraic manipulation and identities to simplify.

• Consider all possible solutions within the specified domain (often ).

• Check for extraneous solutions, especially when squaring both sides.

Reference Angles and Quadrants

Determining Signs of Trigonometric Functions

The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies.

• Quadrant I: All functions positive

• Quadrant II: Sine positive

• Quadrant III: Tangent positive

• Quadrant IV: Cosine positive

Example: If and , then is in Quadrant IV.

Table: Exact Values of Trigonometric Functions at Special Angles

Additional info: Table values inferred from standard trigonometric tables for completeness.