Overview
This lecture introduces the six circular functions (trigonometric functions) and fundamental identities, including their definitions, relationships, domains, ranges, and practical applications.
The Six Circular Functions
- The six circular functions are cosine, sine, secant, cosecant, tangent, and cotangent.
- On the unit circle with point P(x, y):
- cos(θ) = x, sin(θ) = y
- sec(θ) = 1/x if x ≠0, csc(θ) = 1/y if y ≠0
- tan(θ) = y/x if x ≠0, cot(θ) = x/y if y ≠0
- Cosine and sine are defined for all angles; the others have restricted domains.
Reciprocal and Quotient Identities
- sec(θ) = 1/cos(θ); undefined if cos(θ) = 0
- csc(θ) = 1/sin(θ); undefined if sin(θ) = 0
- tan(θ) = sin(θ)/cos(θ); undefined if cos(θ) = 0
- cot(θ) = cos(θ)/sin(θ); undefined if sin(θ) = 0
Reference Angles and Signs
- All circular functions for any angle are equal in magnitude to their value at the reference angle, adjusted by sign depending on the quadrant.
Pythagorean Identities
- cos²(θ) + sin²(θ) = 1
- 1 − sin²(θ) = cos²(θ)
- 1 − cos²(θ) = sin²(θ)
- 1 + tan²(θ) = sec²(θ) (if cos(θ) ≠0)
- 1 + cot²(θ) = csc²(θ) (if sin(θ) ≠0)
Strategies for Verifying Identities
- Start with the more complicated side of an identity.
- Use reciprocal, quotient, and Pythagorean identities.
- Obtain common denominators for rational expressions.
- Use Pythagorean conjugates, e.g., (1 − cos(θ))(1 + cos(θ)) = 1 − cos²(θ).
Circular Functions Beyond the Unit Circle
- For a circle of radius r and point Q(x, y):
- sec(θ) = r/x, csc(θ) = r/y, tan(θ) = y/x, cot(θ) = x/y
Applications and Problem Solving
- Use triangles and the tangent function to solve for heights and distances using angles of inclination or depression.
- In right triangles:
- tan(θ) = opposite/adjacent, sec(θ) = hypotenuse/adjacent, csc(θ) = hypotenuse/opposite, cot(θ) = adjacent/opposite
Domains and Ranges
- cos(θ), sin(θ): domain (−∞, ∞), range [−1, 1]
- sec(θ), csc(θ): domain excludes zeros of cos(θ) or sin(θ); range (−∞, −1] ∪ [1, ∞)
- tan(θ), cot(θ): domain excludes where denominator is zero; range (−∞, ∞)
Key Terms & Definitions
- Circular Functions — Another term for trigonometric functions based on the unit circle.
- Reference Angle — The acute angle formed by the terminal side of an angle and the x-axis.
- Pythagorean Conjugates — Pairs like (1 − cos(θ)), (1 + cos(θ)) whose product simplifies via identities.
Action Items / Next Steps
- Memorize all six circular function definitions and the core identities.
- Practice verifying trigonometric identities as illustrated.
- Complete the assigned exercises at the end of the section.
- Review concepts of reference angles and cofunction identities.