Trigonometric Functions Overview

Trigonometric Functions

• Trigonometric functions relate angles in a right-angled triangle to ratios of its sides.
• The fundamental trigonometric functions are sine (sin), cosine (cos), and tangent (tan).
• Their reciprocals are cosecant (csc), secant (sec), and cotangent (cot), respectively.
• The sine of an angle (sin θ) is the ratio of the side opposite to the angle to the hypotenuse.
• The cosine of an angle (cos θ) is the ratio of the side adjacent to the angle to the hypotenuse.
• The tangent of an angle (tan θ) is the ratio of the side opposite to the angle to the side adjacent to the angle.
• These functions are defined for angles in a right-angled triangle.
• Trigonometric functions can also be used to describe angles in a unit circle.
• The unit circle relates angles to points (x, y) on the circle with radius 1.
• In the unit circle, sin θ represents the y-coordinate and cos θ represents the x-coordinate of the point corresponding to the angle θ.

Trigonometric Identities

• Trigonometric identities are equations that are true for all valid values of the variables.
• Common trigonometric identities include:
  • sin²θ + cos²θ = 1
  • tan²θ + 1 = sec²θ
  • 1 + cot²θ = csc²θ
• Pythagorean identities are fundamental relationships between trigonometric functions.
• These identities are used to simplify expressions and solve trigonometric equations.
• Other identities involve sums and differences of angles, such as
  • sin(A + B) = sinA cosB + cosA sinB
  • cos(A + B) = cosA cosB – sinA sinB
• These identities are used to find the trigonometric values of sums and differences of angles.

Solving Trigonometric Equations

• Trigonometric equations involve trigonometric functions of a variable.
• Solving these involves finding the values of the variable that make the equation true.
• Techniques include using trigonometric identities, factoring, and inverse trigonometric functions.
• Solutions often involve multiple angles due to the periodic nature of trigonometric functions.
• Solutions must be within the specified range of angles or periodic range.

Graphs of Trigonometric Functions

• The graphs of sine, cosine, tangent, and other trigonometric functions are periodic.
• The graph of sin(x), for example, oscillates between -1 and 1 in a repeating pattern.
• The graphs of trigonometric functions are useful for visualizing their behavior and finding solutions to trigonometric equations.
• The period of sin(x) and cos(x) is 2π.
• The period of tan(x) is π.
• Graphs can be shifted horizontally (phase shifts) and vertically.

Inverse Trigonometric Functions

• Inverse trigonometric functions, like arcsin, arccos, and arctan, are used to find the angle given the ratio of sides.
• These functions are used to 'undo' the effect of the trigonometric function.
• Important to note the restricted ranges for these inverse functions. For example, arccos(x) is restricted to the range 0 to π.
• The domains and ranges of inverse trigonometric functions are different from the original functions.