Expert: Sherman D. - 1/1/2007

Question

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The text above is a follow-up to ...

-----Question-----
I had a long trig review packet to do over Christmas break. I have completed most of the questions and was wondering if you could please confirm that some of my answers are correct.

If it is possible to express answers in exact form, you are expected to do so. If exact answers are not possible, round to the nearest tenth of a degree or nearest hundredth of a radian (whichever unit measure the question calls for).

1) Q: Solve for all values of x: sinx=-1 (in radians).
  A: x= 3pi/2 + n times 2pi (Let n= an integer)

2) Q: Find the exact value of tan 105 degrees.
  A: -2 + square root of 3

3) Q: If pi/2 < x < pi and sinx=5/13, find sin 2x.
  A: 120/169 (I tried to reduce it and can't)

4) Q: Find the exact value of csc 300 degrees.
  A: -2 square root of 3 divided by 3

5) Q: Find the exact value of tan(Cos^-1(-square root of 3 divided by 2)).
  A: -square root of 3 divided by three

11)Q: If tan A=2 and tan B= 1/4, find tan(A-B).
  A: 7/6    or    1 and 1/6 (I'm not sure which is more simple)

12) Q: Find the exact value: cos (5pi/12) cos (pi/6) + sin (5pi/12) sin (pi/6)
   A: Square root of 2 divided by 2
Thank you so much for all of your help!!
-----Answer-----
1) Q: Solve for all values of x: sinx=-1 (in radians).
 A: x= 3pi/2 + n times 2pi (Let n= an integer)

sin(x) = -1
x = sin^-1(-1)
x = (3pi/2)

Your answer is correct. Also you should really () what is being multiplied. For ex: until checking your answer, i thought you meant ((3pi/2) + n)(2pi) and not (3pi/2) + (2n)pi

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2) Q: Find the exact value of tan 105 degrees.
 A: -2 + square root of 3

tan(105) =
tan(45 + 60) =
(tan(45) + tan(60))/(1 - tan(45)tan(60)) =
(1 + sqrt(3))/(1 - sqrt(3))
((1 + sqrt(3))(1 + sqrt(3)))/(-2)
(1 + sqrt(3) + sqrt(3) + 3)/(-2)
(4 + 2sqrt(3))/(-2)
-2 - sqrt(3)

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3) Q: If pi/2 < x < pi and sinx=5/13, find sin 2x.
 A: 120/169 (I tried to reduce it and can't)

sin(x) = (5/13)

5^2 + y^2 = 13^2
25 + y^2 = 169
y^2 = 144
y = -12 or 12

cos(x) = (12/13)

sin(2x) = 2sin(x)cos(x) = 2(5/13)(12/13) = (120/169)

Thats what i got.

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4) Q: Find the exact value of csc 300 degrees.
 A: -2 square root of 3 divided by 3

csc(300) =
1/(sin(300)) =
1/(sin(360 - 60)) =
1/(sin(360)cos(60) - sin(60)cos(360)) =
1/(-sqrt(3)/2) =
-2/(sqrt(3)) =
(-2sqrt(3))/3
Correct

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5) Q: Find the exact value of tan(Cos^-1(-square root of 3 divided by 2)).
 A: -square root of 3 divided by three

tan(cos^-1(-sqrt(3)/2))) =
tan(150) =
tan(180 - 30) =
(tan(180) - tan(30))/(1 + tan(180)tan(30))
(0 - (sqrt(3)/3))/1
(-sqrt(3)/3)
Correct

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11)Q: If tan A=2 and tan B= 1/4, find tan(A-B).
 A: 7/6    or    1 and 1/6 (I'm not sure which is more simple)

tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B))

(2 - (1/4))/(1 + (2 * (1/4)))
((8 - 1)/4)/(1 + (1/2))
(7/4)/((2 + 1)/2)
(7/4)/(3/2)
(7/4)*(2/3)
(14/12)
(7/6)
Correct

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12) Q: Find the exact value: cos (5pi/12) cos (pi/6) + sin (5pi/12) sin (pi/6)
  A: Square root of 2 divided by 2

cos(5pi/12)cos(pi/6) + sin(5pi/12)sin(pi/6) =
cos((5pi/12) - (pi/6)) =
cos((5pi/12) - (2pi/12)) =
cos(3pi/12) =
cos(pi/4) =
(sqrt(2)/2)
Correct

Well except for #2, you got everything else correct.

I really appreciate everything that you have done for me already bit could you just clarify something from question #2: in the end, how you went from (4 + 2 square root of 3)/(-2) to -2-square root of 3. i trust your answer but i don't understand how you get the negative. i am probably just missing something really obvious... Thanks SOOOOOOOOOO much!!! :)

Answer
tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B))

tan(105) = tan(45 + 60)

tan(45 + 60) = (tan(45) + tan(60))/(1 - tan(45)tan(60)) =
(1 + sqrt(3))/(1 - (1 * sqrt(3))) = (1 + sqrt(3))/(1 - sqrt(3))

Multiply top and bottom by (1 + sqrt(3)) and you get

((1 + sqrt(3))(1 + sqrt(3)))/((1 - sqrt(3))(1 + sqrt(3)))
(1 + sqrt(3) + sqrt(3) + 3)/(1 + sqrt(3) - sqrt(3) - 3)
(4 + 2sqrt(3))/(1 - 3)
(4 + 2sqrt(3))/(-2)

since both the 4 and (2sqrt(3)) are divided by the -2, you get

-2 - sqrt(3) instead of -2 + sqrt(3), if you were to say tan(105) = -2 - sqrt(3), you get that both sides are equal. However -2 + sqrt(3) gets you -.26795 and tan(105) gets you -3.732

Info found at www.math.com/tables/trig/identities.htm