Results 1 to 2 of 2

Math Help - Trigonometry Help (Identities)

  1. #1
    Junior Member
    Joined
    Jun 2006
    Posts
    37

    Trigonometry Help (Identities)

    Hi!
    I just have one sum I need.
    xsinθ + ycosθ = sinθ.cosθ AND xsinθ - ycosθ = 0

    Prove that: x + y = 1

    I tried a lot, but couldn't get it.

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,914
    Thanks
    779
    Hello, Ruler of Hell!

    Given: . \begin{array}{cccc}x\sin^3\!\theta + y\cos^3\!\theta \:= \:\sin\theta\cos\theta & {\color{blue}[1]}\\ x\sin\theta - y\cos\theta \:=\:0 & {\color{blue}[2]}\end{array}

    Prove that: . x^2 + y^2 \:=\: 1

    From [2], we have: . y\cos\theta \,=\,x\sin\theta \quad\Rightarrow\quad y \,=\,\frac{x\sin\theta}{\cos\theta} .[3]

    Substitute into [1]: . x\sin^3\!\theta + \left(\frac{x\sin\theta}{\cos\theta}\right)\cos^3\  !\theta \:=\:\sin\theta\cos\theta

    . . . . . . . x\sin^3\!\theta + x\sin\theta\cos^2\!\theta \:=\:\sin\theta\cos\theta


    \text{Factor: }\;x\sin\theta\underbrace{(\sin^2\!\theta + \cos^2\!\theta)}_{\text{This is 1}} \:=\:\sin\theta\cos\theta

    . . . . . . x\sin\theta \:=\:\sin\theta\cos\theta \quad\Rightarrow\quad\boxed{ x \:=\:\cos\theta}


    Substitute into [3]: . y \:=\:\frac{(\cos\theta)(\sin\theta)}{\cos\theta} \quad\Rightarrow\quad\boxed{ y \:=\:\sin\theta}


    Therefore: . x^2 + y^2 \;=\;\cos^2\!\theta + \sin^2\!\theta \;=\;1

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Trigonometry Identities? Please Help.
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: December 7th 2010, 09:35 AM
  2. Trigonometry help with identities
    Posted in the Trigonometry Forum
    Replies: 4
    Last Post: July 10th 2010, 01:40 PM
  3. Trigonometry Identities
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: April 30th 2009, 04:39 AM
  4. Trigonometry Identities
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: April 29th 2009, 08:20 PM
  5. Trigonometry Identities
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: June 18th 2008, 03:25 PM

Search Tags


/mathhelpforum @mathhelpforum