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1. Evaluate $\tan(\theta )\,\$ in terms of $\sin(\theta )\,\$ $\tan(\theta )=\sin(\theta )/{\sqrt {(}}1-(\sin ^{2}(\theta ))\,\$ Shyam (T/C)
2. If $\csc(\theta )=1/x,\,\$ then what does $x\,\$ equal?
$x=\sin(\theta )\,\$ where $x=[-1,1]\,\$ Shyam (T/C)
3. Prove $\tan ^{2}(\theta )+1=\sec ^{2}(\theta )\,\$ using $\,\ \sin ^{2}(\theta )+\cos ^{2}(\theta )=1$ $\sin ^{2}(\theta )+cos^{2}(\theta )=1\,\$ divide both sides by $\cos ^{2}(\theta )=>\sin ^{2}(\theta )/cos^{2}(\theta )+1=1/cos^{2}(\theta )\,\$ $=>\tan ^{2}(\theta )+1=sec^{2}(\theta )\,\$ Shyam (T/C)
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)\,\$ Shyam (T/C)
• 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\,\$ Shyam (T/C)
• 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\,\$ Shyam (T/C)
• (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)-2sin^{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\,\$ Shyam (T/C) 19:42, 18 November 2006 (UTC)