📚Study Guide: Trigonometric and Polar Functions
Unit 3: Trigonometric and Polar Functions
This unit explores the rich world of trigonometric functions and their applications, along with an introduction to polar coordinates and functions. You will master the unit circle, radian measure, and the graphs of sine, cosine, tangent, and their reciprocals. Inverse trigonometric functions and their restricted domains are essential for solving equations. Trigonometric identities--Pythagorean, sum and difference, double-angle, and half-angle--enable simplification and proof. You will also solve trigonometric equations and apply the Law of Sines and Law of Cosines to non-right triangles. Polar coordinates provide an alternative way to locate points and graph functions, setting the stage for calculus applications.
Key Concepts
- Unit Circle: Defines sine and cosine for all real numbers; memorize quadrantal angles and reference angles.
- Radian Measure: One radian is the angle subtended by an arc equal in length to the radius. 180 degrees = pi radians.
- Graphs of Sine and Cosine: Amplitude = |A|, period = 2pi/|B|, phase shift = -C/B, vertical shift = D for y = A*sin(B(x-C)) + D.
- Inverse Trig Functions: arcsin, arccos, arctan have restricted domains and ranges to ensure they are functions.
- Pythagorean Identities: sin^2(x) + cos^2(x) = 1; 1 + tan^2(x) = sec^2(x); 1 + cot^2(x) = csc^2(x).
- Sum and Difference Formulas: Expand sin(A +/- B) and cos(A +/- B).
- Solving Trig Equations: Use identities, factoring, and the unit circle to find all solutions within a specified interval.
- Polar Coordinates: A point is described by (r, theta); conversion formulas link polar and rectangular systems.
Vocabulary
- Amplitude: Half the distance between the maximum and minimum values of a sinusoidal function.
- Period: The length of one complete cycle of a periodic function.
- Phase shift: A horizontal displacement of a trigonometric graph from its standard position.
- Reference angle: The acute angle formed by the terminal side and the x-axis.
- Coterminal angles: Angles that share the same terminal side but differ by full rotations.
- Polar axis: The ray from the pole (origin) representing the positive x-axis in the polar coordinate system.
Formulas
- sin^2(x) + cos^2(x) = 1
- 1 + tan^2(x) = sec^2(x)
- sin(A +/- B) = sin(A)cos(B) +/- cos(A)sin(B)
- cos(A +/- B) = cos(A)cos(B) -/+ sin(A)sin(B)
- sin(2A) = 2sin(A)cos(A)
- cos(2A) = cos^2(A) - sin^2(A) = 2cos^2(A) - 1 = 1 - 2sin^2(A)
- Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
- Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C)
- Polar to rectangular: x = r*cos(theta), y = r*sin(theta)
Common Mistakes
- Using degrees instead of radians in calculus contexts; always verify the mode setting.
- Forgetting that inverse sine and cosine have restricted ranges, leading to extraneous solutions.
- Confusing the period formula for tangent (pi/|B|) with sine and cosine (2pi/|B|).
- Missing negative solutions or additional coterminal solutions when solving trigonometric equations.
AP Exam Strategies
- Always draw the unit circle or a reference triangle when evaluating trig functions of special angles.
- For graphing transformations, identify amplitude, period, phase shift, and vertical shift before plotting key points.
- When proving identities, work on one side only, converting everything to sine and cosine if stuck.
- Use the Law of Cosines for SAS or SSS triangle configurations; use Law of Sines for AAS or ASA, and watch for the ambiguous case.
Real-World Applications
- Physics: Simple harmonic motion, wave behavior, and alternating current are modeled with sinusoidal functions.
- Navigation: The Law of Cosines calculates distances and bearings in GPS and aviation.
- Music: Sound waves are periodic functions; pitch corresponds to frequency (related to period).