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Trigonometric Identities quiz

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  • What are the three criteria for a fully simplified trigonometric expression?
    All arguments must be positive, the expression should contain no fractions, and it should have as few trig functions as possible.
  • Which identity allows you to rewrite tan(-θ) as a function of tan(θ)?
    The even-odd identity states that tan(-θ) = -tan(θ).
  • How do you convert tangent and cosecant to sine and cosine?
    Tangent is sine over cosine (tan(θ) = sin(θ)/cos(θ)), and cosecant is 1 over sine (csc(θ) = 1/sin(θ)).
  • What is the simplified form of tan(-θ) * csc(θ)?
    It simplifies to -sec(θ) by applying identities and reducing to a single trig function.
  • What strategy should you use when you see 1 plus or minus a trig function in a fraction's denominator?
    Multiply both the numerator and denominator by 1 minus or plus that same trig function to use the difference of squares.
  • How does sin²(θ) / (1 + cos(θ)) simplify?
    It simplifies to 1 - cos(θ) by multiplying by (1 - cos(θ)) and using the Pythagorean identity.
  • What is the Pythagorean identity involving sine and cosine?
    sin²(θ) + cos²(θ) = 1.
  • What is the process for verifying a trigonometric identity?
    Simplify one or both sides of the equation using identities until both sides are equal.
  • How do you simplify sin(θ) * cos(θ) / (1 - cos²(θ))?
    Replace 1 - cos²(θ) with sin²(θ), then simplify to cot(θ) or 1/tan(θ).
  • What identity relates sec²(θ) and tan²(θ)?
    tan²(θ) + 1 = sec²(θ), so sec²(θ) - tan²(θ) = 1.
  • How do you handle cosine of a negative angle, cos(-θ)?
    Cosine is an even function, so cos(-θ) = cos(θ).
  • What is the result of simplifying sec²(θ) - tan²(θ) / (cos(-θ) + 1)?
    It simplifies to 1 / (cos(θ) + 1) by using identities and even-odd properties.
  • What should you do if you get stuck while simplifying or verifying a trig identity?
    Start over from scratch, as a new approach may reveal a solution.
  • How do you solve sec²(θ) - 1 / tan(θ) = 1?
    Rewrite sec²(θ) - 1 as tan²(θ), simplify to tan(θ) = 1, and solve for θ using the unit circle.
  • How do you use the double angle identity to simplify sin(2θ) / cos(-θ) = 1?
    Rewrite sin(2θ) as 2sin(θ)cos(θ), use cos(-θ) = cos(θ), cancel cos(θ), and solve 2sin(θ) = 1 for θ.