trig function questions (1 Viewer)

[Schniz]

1)
Simplify in terms of the indicated angle:
1-2sin^2 3A; in terms of 6A

2)
Simplify:
cosA(cos4A+2sin^2 2A)

3)
Prove:
(cos@+sin@)/(cos@-sin@) = sec2@+tan2@

4)
Write down expressions for tan3@ and cos3@ in terms of the angle @. (argh what kind of question i sthis??? :S)

5)
Evaluate the integral from pi/4 to 0 of (cos^2 @ -1)

6)
Write down a formula for sin3@ in terms of the angle @



thanks... any help is muchly appreciated

 

[SoulSearcher]

i) 1-2sin²3A = cos6A
ii) cosA(cos4A+2sin²2A) =
cosA(1-2sin²2A+2sin²2A)
= cosA
iii) LHS = (cos@+sin@)/(cos@-sin@)
= (cos@+sin@)/(cos@-sin@) * (cos@+sin@)/(cos@+sin@)
= (cos²@+sin²@+2sin@cos@)/{cos²@-sin²@)
= (1+sin2@)/cos2@
= sec2@+tan2@
= RHS
iv) tan3@ = tan(2@+@)
= (tan2@+tan@)/(1-tan@tan2@)
= {[2tan@/(1-tan²@)] + tan@}/(1-tan@[2tan@/(1-tan²@)])
= (3tan@-tan³@/1-tan²@)/(1-3tan²@/1-tan²@)
= 3tan@-tan³@/1-3tan²@
cos3@ = cos(2@+@)
= cos2@cos@ - sin2@sin@
= 2cos³@ - cos^{@ - (2cos@ - 2cos3@)
= 4cos3@ - 3cos@}

 

[A l]

5) 1 if it is 0 to pi/4, -1 if it is actually pi/4 to 0

 

[Schniz]

thanks... im not sure if i understand how you did the first one... im fairly dumb...

 

[ianc]

Question 5: (i did not get AI's answer)

cos²@-1 = 1/2 + 1/2(cos2@)-1
= -1/2 + 1/2(cos2@)

Hence the integral of cos²@-1 =
integral -1/2 + 1/2(cos2@)
= [-@/2 + 1/4sin2@] + c

then substituting in the limits of pi/4 and 0:
= -pi/8 + 1/4
= -0.143 (3 decimal places)

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Question 6: (just using the double angle formula and the identity cos²@+sin²@=1)

sin3@
= sin@cos2@ + cos@sin2@
= sin@(cos²@ - sin²@) + cos@(2sin@cos@)
= sin@cos²@ - sin@³@ + 2cos²@sin@
= sin@(1 - sin²@) - sin@³@ + 2sin@(1 - sin²@)
= 3sin@ - 4sin³@

 

[ianc]

for the first question, you use the double angle formula for cos2A

fully worked out:
1 - 2sin²3A
= 1 - sin²3A - sin²3A
= 1 - (1-cos²3A) - sin²3A
= cos²3A - sin²3A
= cos[2(3A)]
=cos6A

hope this helps!

 

[SoulSearcher]

vi) sin 3@ = sin (2@+@)
= sin2@cos@ + cos2@sin@
= 2sin@cos²@ + (1-2sin²@)sin@
= 2sin@(1-sin²@) + (1-2sin²@)sin@
= 3sin@ - 4sin³@

 

[Schniz]

thanks everyone