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Trigonometry and Complex Numbers Flashcards

6 cards from real AIME practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

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  1. What is sin^2(θ) + cos^2(θ) for any angle θ?

    Answer: 1

    This is the fundamental Pythagorean identity, which equals 1 for all θ.

  2. Using the Law of Cosines, find side c when a=5, b=7, C=60°.

    Answer: √39

    c^2=25+49-2(5)(7)(1/2)=74-35=39, so c=√39.

  3. Express sin(2θ) using a double-angle formula.

    Answer: 2sin(θ)cos(θ)

    The double-angle identity is sin(2θ)=2sin(θ)cos(θ).

  4. What is the argument (angle) of the complex number z = -1 + i?

    Answer: 135°

    -1+i lies in the second quadrant; reference angle=arctan(1/1)=45°, so argument=180°-45°=135°.

  5. Compute |3 - 4i|^2.

    Answer: 25

    |3-4i|=√(9+16)=5; |3-4i|^2=25.

  6. How many solutions does 2cos(x) = 1 have in the interval [0, 2π)?

    Answer: 2

    cos(x)=1/2 gives x=π/3 and x=5π/3 in [0,2π), so there are 2 solutions.