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Introduction to Trigonometric Identities quiz

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  • Which trigonometric functions are even and which are odd?
    Cosine and secant are even functions; sine, tangent, cosecant, and cotangent are odd functions.
  • What does it mean for a function to be even?
    An even function satisfies f(-x) = f(x) and is symmetric about the y-axis.
  • What does it mean for a function to be odd?
    An odd function satisfies f(-x) = -f(x) and is symmetric about the origin.
  • What is the even-odd identity for cosine?
    cos(-θ) = cos(θ)
  • What is the even-odd identity for sine?
    sin(-θ) = -sin(θ)
  • What is the even-odd identity for tangent?
    tan(-θ) = -tan(θ)
  • State the basic Pythagorean identity involving sine and cosine.
    sin²θ + cos²θ = 1
  • How do you derive the identity tan²θ + 1 = sec²θ?
    Divide the basic Pythagorean identity by cos²θ.
  • How do you derive the identity 1 + cot²θ = csc²θ?
    Divide the basic Pythagorean identity by sin²θ.
  • When should you use even-odd identities in simplifying trigonometric expressions?
    Use even-odd identities whenever the argument of a trig function is negative.
  • What are the three criteria for a trigonometric expression to be fully simplified?
    All arguments are positive, there are no fractions, and there are as few trig functions as possible.
  • What strategy can you use if your expression contains 1 plus or minus a trig function in the denominator?
    Multiply the numerator and denominator by the conjugate (1 minus or plus the trig function) to simplify.
  • How do you simplify tan(-θ)·csc(θ)?
    Rewrite tan(-θ) as -tan(θ), express in terms of sine and cosine, and simplify to -sec(θ).
  • How do you verify a trigonometric identity?
    Simplify one or both sides of the equation using identities and algebraic manipulation until both sides are equal.
  • What is the result of sec²θ - tan²θ?
    sec²θ - tan²θ = 1, by rearranging the Pythagorean identity tan²θ + 1 = sec²θ.