Introduction to Calculus/Quiz 1/Answers

If you can pass this quiz, you are ready to take this course

1. Evaluate $\tan (\theta )$ in terms of $\sin (\theta )$

   $\tan (\theta )=\sin (\theta )/\sqrt{(}1-({\sin }^2(\theta ))$

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2. If $\csc (\theta )=1/x,$ then what does $x$ equal?

   $x=\sin (\theta )$ where $x=[-1,1]$

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3. Prove ${\tan }^2(\theta )+1={\sec }^2(\theta )$using ${\sin }^2(\theta )+{\cos }^2(\theta )=1$

   ${\sin }^2(\theta )+co{s}^2(\theta )=1$

   divide both sides by ${\cos }^2(\theta )=>{\sin }^2(\theta )/co{s}^2(\theta )+1=1/co{s}^2(\theta )$

   $=>{\tan }^2(\theta )+1=se{c}^2(\theta )$

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4. $\cos (A+B)=\cos (A)\cos (B)-\sin (A)\sin (B)$
   • Find the double angle idenities for the cosine function using the above rule.

     replace $B$ by $A=>\cos (A+A)=\cos (A)\cos (A)-\sin (A)\sin (A)$

     $=>\cos (2A)={\cos }^2(A)-{\sin }^2(A)$

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   • Find the half angle idenities from the double angle idenities.

     $=>\cos (2A)={\cos }^2(A)-{\sin }^2(A)$

     replace $A$ by $A/2=>\cos (A)=2{\cos }^2(A/2)-1$ using ${\sin }^2(A)+{\cos }^2(A)=1$

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   • Find the value of ${\cos }^2(\theta )$ without exponents using the above rules

     $=>\cos (2\theta )=2{\cos }^2(\theta )-1$

     $=>{\cos }^2(\theta )=(1+\cos (2\theta ))/2$

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   • (Challenge) Find the value of ${\cos }^3(\theta )$ without exponents

     $\cos (3\theta )=cos(\theta +2\theta )$

     $=>\cos (3\theta )=cos(\theta )\cos (2\theta )-sin(\theta )\sin (2\theta )$

     $=>\cos (3\theta )=cos(\theta )(2{\cos }^2(\theta )-1)-sin(\theta )(2\sin (\theta )\cos (\theta ))$ using $\cos (2\theta )=2{\cos }^2(\theta )-1$ and $\sin (2\theta )=2\sin (\theta )\cos (\theta )$

     $=>\cos (3\theta )=cos(\theta )((2{\cos }^2(\theta )-1)-2si{n}^2(\theta ))$

     $=>\cos (3\theta )=cos(\theta )(4{\cos }^2(\theta )-3)$ using ${\sin }^2(\theta )+{\cos }^2(\theta )=1$

     $=>4{\cos }^3(\theta )=cos(3\theta )+3cos(\theta )$

     $=>{\cos }^3(\theta )=(cos(3\theta )+3cos(\theta ))/4$

   [[User:Shyam| Template:Font]] ^{([[User talk:Shyam|Template:Font]]/[[Special:Contributions/Shyam|Template:Font]])} 19:42, 18 November 2006 (UTC)