BackPrecalculus: Trigonometric Identities and Verification Techniques
6:36
Terms in this set (21)
sin θ = tan θ · cos θ and cos θ = cot θ · sin θ
csc θ = 1 / sin θ, sec θ = 1 / cos θ, cot θ = 1 / tan θ
sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, 1 + cot² θ = csc² θ
sin(-θ) = -sin θ, tan(-θ) = -tan θ, csc(-θ) = -csc θ, cot(-θ) = -cot θ
cos(-θ) = cos θ, sec(-θ) = sec θ
a² - b² = (a + b)(a - b)
(a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b²
Use reciprocal, quotient, or Pythagorean identities to rewrite expressions and verify identities.
Rewrite all trigonometric functions in terms of sin θ and cos θ to simplify verification.
Use algebraic factoring such as greatest common factor, difference of squares, or perfect squares to simplify expressions.
Rewrite \(\frac{A+B}{C}\) as \(\frac{A}{C}+\frac{B}{C}\) to simplify.
Use a common denominator to combine fractions: \(\frac{A}{B}+\frac{C}{D}=\frac{AD+BC}{BD}\).
Multiply numerator and denominator by the conjugate \(A-B\) to simplify expressions of the form \(\frac{A+B}{C}\).
Start with the more complex side and try to transform it to match the other side using known identities and algebraic techniques.
Use variations of Pythagorean identities when the expression includes squared trigonometric functions.
Use even/odd properties to simplify expressions involving negative angles.
cot θ = 1 / tan θ
sec θ = 1 / cos θ
csc θ = 1 / sin θ
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ