Chapter 7: Analytic Trigonometry
Analytic trigonometry develops the algebraic side of trig — simplifying expressions using identities, solving trig equations, and deriving powerful formulas like the sum, difference, double-angle, and half-angle formulas. These tools form the backbone of calculus and physics.
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Sections
7.1Trigonometric Identities
An identity is an equation true for ALL values of the variable. We build our toolkit starting from the Pythagorean identities and learn strategies for proving and simplifying trig expressions.
7.2Addition and Subtraction Formulas
The sum and difference formulas let you compute exact trig values for angles like 75° or 15° by splitting them into known angles (30°, 45°, 60°, 90°). They also enable algebraic manipulation of trig expressions.
7.3Double-Angle, Half-Angle, and Product-Sum Formulas
These formulas extend the addition formulas to special cases. Double-angle formulas are essential in calculus. Half-angle formulas give exact values for 22.5°, 15°, etc. Power-reducing formulas convert squares to double angles, enabling integration.
7.4Inverse Trigonometric Functions
Inverse trig functions answer 'which angle has this trig value?' They have restricted domains and ranges to ensure they are true functions. Mastering domain and range restrictions, exact value evaluation, and compositions is essential for calculus.
7.5Trigonometric Equations
Trig equations have infinitely many solutions because trig functions are periodic. We find all solutions in one period then add multiples of the period. More complex equations require combining identities, factoring, or substitution before solving.
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