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Solving Trigonometric Equations Using Identities quiz

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  • What is the main goal when solving trigonometric equations?
    The main goal is to find an angle theta that makes the equation true.
  • Why do we use trigonometric identities when solving equations with multiple trig functions?
    We use identities to rewrite the equation in terms of just one trig function, making it easier to solve.
  • Which identity can be used to rewrite sec^2(theta) - 1?
    The Pythagorean identity can be used, and sec^2(theta) - 1 equals tan^2(theta).
  • After simplifying sec^2(theta) - 1 over tan(theta), what does the equation become?
    It becomes tan(theta) = 1.
  • How do you find all solutions to tan(theta) = 1?
    Find the angles where tan(theta) = 1 on the unit circle and add pi n to account for all solutions.
  • Why can't you directly simplify sin(2theta)/cos(-theta) to tan(2theta)?
    Because the arguments of the sine and cosine functions are not the same.
  • Which identity is used to rewrite sin(2theta)?
    The double angle identity: sin(2theta) = 2 sin(theta) cos(theta).
  • How do you simplify cos(-theta) using an identity?
    Use the even-odd identity; cos(-theta) = cos(theta) because cosine is an even function.
  • What happens to 2 sin(theta) cos(theta) / cos(theta) after simplification?
    The cos(theta) terms cancel, leaving 2 sin(theta) = 1.
  • How do you isolate sin(theta) in the equation 2 sin(theta) = 1?
    Divide both sides by 2 to get sin(theta) = 1/2.
  • Which angles on the unit circle have sin(theta) = 1/2?
    Theta = pi/6 and theta = 5pi/6.
  • How do you express all solutions for sin(theta) = 1/2?
    Add 2pi n to each solution: theta = pi/6 + 2pi n and theta = 5pi/6 + 2pi n.
  • What is the purpose of adding 2pi n or pi n to solutions?
    It accounts for all possible solutions by including all coterminal angles.
  • Why is it important to recognize when to use double angle or even-odd identities?
    Recognizing these identities helps simplify equations with different arguments or negative angles.
  • What is the general strategy for solving complex trigonometric equations?
    Identify and apply appropriate identities to reduce the equation to a single trig function, then solve for theta.