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Math Help - a few problems having trouble with

  1. #1
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    a few problems having trouble with

    okay I have like 50 problems to do and I have a few I need help with particularly. If someone can help, I would appreciate it so much.

    Problem 1:
    Prove the identity.

    tan(t+(pi/2))=-cot(t).


    Problem 2:
    Rewrite sin(5t)-sin(3t).


    Problem 3:
    Solve 3 cos(theta)+3=2 sin^2(theta) where 0<=theta<2pi


    Problem 4:
    Solve cos(theta)+sin(theta)=1 where 0<=theta<2pi
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  2. #2
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    Hello, lsnyder!

    Here's some help . . .


    (3) Solve: . 3\cos\theta+3\:=\:2\sin^2\!\theta\:\text{ where }0 \leq\theta < 2\pi
    Replace \sin^2\!\theta with 1-\cos^2\!\theta

    3\cos\theta + 3 \:=\:2(1-\cos^2\!\theta) \quad\Rightarrow\quad 3\cos\theta \:=\:2-2\cos^2\!\theta

    . . 2\cos^2\!\theta + 3\cos\theta + 1 \:=\:0 \quad\Rightarrow\quad (\cos\theta + 1)(2\cos\theta+1) \:=\:0


    Then: . \begin{array}{cccccccccccc}<br />
\cos\theta + 1 \:=\:0 & \Rightarrow & \cos\theta \:=\:\text{-}1 & \Rightarrow & \theta \:=\:\pi \\<br />
2\cos\theta + 1 \:=\:0 & \Rightarrow & \cos\theta \:=\:\text{-}\frac{1}{2} & \Rightarrow & \theta \:=\:\frac{2\pi}{3},\:\frac{4\pi}{3}\end{array}




    (4) Solve: . \cos\theta+\sin\theta \:=\:1\:\text{ where }0 \leq \theta < 2\pi

    Square both sides: . (\cos\theta + \sin\theta)^2 \:=\:1^2 \quad\Rightarrow\quad \cos^2\!\theta + 2\cos\theta\sin\theta + \sin^2\!\theta \:=\:1

    . . \underbrace{2\sin\theta\cos\theta}_{\text{This is }\sin2\theta} + \underbrace{\sin^2\!\theta + \cos^2\!\theta}_{\text{This is 1}} \:=\:1 \quad\Rightarrow\quad \sin2\theta + 1 \:=\:1 \quad\Rightarrow\quad \sin2\theta \:=\:0

    . . 2\theta \:=\:0,\:\pi,\:2\pi,\;3\pi \quad\Rightarrow\quad \theta \;=\;0,\:\tfrac{\pi}{2},\;\pi,\;\tfrac{3\pi}{2}


    Since squaring an equation often introduces extraneous roots,
    . . we must check our answers.

    And we find that \theta \:=\:\pi\,\text{ and }\,\tfrac{3\pi}{2} are extraneous.

    Therefore, the answers are: . \theta \;=\;0,\:\frac{\pi}{2}

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  3. #3
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    okay, i got a question for number 1, which probably sounds so dumb but would it start this.....

    tan(t)+tan(pi/2)=-cot (t)
    tan(t)+tan(pi/2)=- 1/tan(t)

    or....

    sin/cos(t+(pi/2))=-cot(t)
    sin/cos(t+(pi/2))=-1/tan(t)

    or is that way off.
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  4. #4
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    Exclamation

    i figured out that number 1 is an cofunction idenity

    so it is set up as

    tan(t+pi/2)=>(tan (t)-tan (pi/2)/(1+tan(t)tan(pi/2)



    after that I am stuck.
    does anyone know what to do
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