Trigonometry Mastery Cheat Sheet

This cheat sheet covers fundamental unit circle values, trigonometric identities, inverse functions, graphing, equation solving, and applications, providing a comprehensive overview of trigonometry for quick reference and study.

Core Principles

• Unit Circle: Key for evaluating trig functions at common angles.
• Identities: Simplify expressions and prove equations.
• Inverse Functions: Understand domain/range for unique outputs.
• Graphing: Visualize periodic behavior and transformations.
• Solving Equations: Find angles that satisfy trig statements.
• Applications: Use trig to solve real-world problems.

Action Steps

• Memorize the Unit Circle: Focus on key angles (0, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$) and their coordinates.
• Practice Identity Verification: Start with the complex side and simplify to the simpler side.
• Understand Inverse Domains/Ranges: Crucial for finding principal values.
• Sketch Parent Graphs: Know the basic shape of all six trig functions.
• Apply Transformations: Systematically adjust amplitude, period, phase, and vertical shift.
• Solve Equations Step-by-Step: Isolate the trig function, find reference angles, and consider all quadrants.
• Draw Triangles for Word Problems: Label sides and angles correctly.
• Check for Ambiguous Case (Law of Sines): If given SSA, determine if 0, 1, or 2 triangles exist.

Formulas

• Reciprocal Identities: $\csc x = \frac{1}{\sin x}$, $\sec x = \frac{1}{\cos x}$, $\cot x = \frac{1}{\tan x}$
• Quotient Identities: $\tan x = \frac{\sin x}{\cos x}$, $\cot x = \frac{\cos x}{\sin x}$
• Pythagorean Identities: $\sin^2 x + \cos^2 x = 1$, $1 + \tan^2 x = \sec^2 x$, $1 + \cot^2 x = \csc^2 x$
• Sum/Difference (Sine): $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• Sum/Difference (Cosine): $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
• Sum/Difference (Tangent): $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
• SOH-CAH-TOA: $\sin \theta = \frac{Opp}{Hyp}$, $\cos \theta = \frac{Adj}{Hyp}$, $\tan \theta = \frac{Opp}{Adj}$
• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$

Key Terms

• Unit Circle: A circle with radius 1 centered at the origin, used to define trigonometric functions for all real numbers.
• Amplitude: Half the distance between the maximum and minimum values of a periodic function (for sine and cosine).
• Period: The smallest interval over which a periodic function completes one full cycle.
• Phase Shift: A horizontal translation of the graph of a trigonometric function.
• Reference Angle: The acute angle formed between the terminal side of an angle and the x-axis.
• Ambiguous Case: Occurs in the Law of Sines when given two sides and a non-included angle (SSA), potentially leading to zero, one, or two triangles.

Pro Tips

• Unit Circle Shortcut: If you forget coordinates, remember the pattern of $\sqrt{0}/2, \sqrt{1}/2, \sqrt{2}/2, \sqrt{3}/2, \sqrt{4}/2$ for sine (y-values) and cosine (x-values) in the first quadrant.
• Identity Proof Strategy: Always rewrite everything in terms of sine and cosine first.
• Graphing Sine/Cosine: Think of the basic 'parent' graph and then stretch/shift it.
• Solving Equations: Always consider the reference angle and then adjust for the correct quadrants based on the sign.
• Ambiguous Case (SSA): Use the Law of Sines. If $h = a \sin A$ and $h < b < a$, there are two solutions.
• Visualize: Draw the unit circle, graphs, and triangles to aid understanding.

Pitfalls to Avoid

• Confusing reciprocal and co-function identities.
• Forgetting the $\pm$ or $\mp$ in sum/difference formulas.
• Incorrectly applying domain/range restrictions for inverse functions.
• Mixing up amplitude, period, and phase shift in graphing.
• Missing solutions when finding general solutions to equations.
• Assuming a unique solution in the SSA case for the Law of Sines.
• Calculation errors with Law of Cosines, especially with signs.

Real World Examples

• Calculating the height of a tree using its shadow length and the angle of elevation.: Right triangle trigonometry (SOH-CAH-TOA).
• Determining the distance between two cities when only their distances from a third point and the angle between those lines are known.: Law of Cosines.
• Modeling sound waves or alternating electrical current.: Graphing sine and cosine functions with transformations.
• Finding the angle needed to hit a target at a specific distance.: Solving trigonometric equations.

Timeline

• Ancient Greece (c. 150 BC): Hipparchus develops trigonometry to calculate astronomical tables.
• c. 100 AD: Ptolemy's Almagest uses trigonometric tables (chords) for astronomy.
• 5th Century AD: Indian mathematicians develop sine tables and use it in astronomy.
• 10th Century AD: Islamic mathematicians refine trigonometry, introduce tangent function.
• 15th Century: European scholars translate Arabic works, 'trigonometry' term coined.
• 17th-18th Centuries: Euler and others develop modern analytic trigonometry and complex analysis connections.

People

• Hipparchus: Often called the 'father of trigonometry', created early trigonometric tables.
• Ptolemy: Included trigonometric tables (chords) in his influential astronomical work, the Almagest.
• Leonhard Euler: Pioneered modern analytic trigonometry, linking it to complex numbers and calculus.