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Advanced Math/Inverse trigonometric functions

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Expert: - 11/16/2010


Question

trig question
The problem is:
sin(2cos^(-1) (-5/7)+11π/12)

I have attached an image of the problem setup. I've tried the question several times and do not have the answer. The only comment I have from the assistant professor is that my answer was not correct. I don't know how to do this problem and it's driving me nuts. I need to see the steps in how to solve it. Any help appreciated. Thank you.
Kimberly

Answer
Questioner:   Kimberly Ivens
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  Trig sin inverse cosine
Question:  

trig questionThe problem is:
sin(2cos^(-1) (-5/7)+11π/12)

I have attached an image of the problem setup. I've tried the question several times and do not have the answer. The only comment I have from the assistant professor is that my answer was not correct. I don't know how to do this problem and it's driving me nuts. I need to see the steps in how to solve it. Any help appreciated. Thank you.
Kimberly  
.........................................

Perhaps you need a full professor (ask him after dinner).

OK, enough with the funny stuff.

This problem will take some work.  No wonder he didn't want to do it.

The key is that an inverse trig expression is an angle.  So write:

T = arccos(-5/7)    [I prefer this notation -- easier to type.]

Then  cos T = -5/7, and you can compute sin T from that:

r = 7,  x = -5,  y = sqrt(24), so

sin T = s24/7   << saving typing.

Now then:

sin(2cos^(-1) (-5/7) + 11π/12) =

sin(2T + 11π/12)

Now use a reduction formula:

sin(A + B) = sin A cos B + cos A sin B

sin(2T + 11π/12) = sin 2T cos 11π/12 + cos 2T sin 11π/12

And another couple:

sin 2T = 2 sin T cos T   << < You compute this.
cos 2T = 2 cos^2 T - 1   << < You compute this, too.

(remember, we know what sin T and cos T are equal to.)
.......................
What about 11pi/12?  We need some work for that, too.  

Remember, the 'special' angles are:

pi/3 = 60 deg,
pi/4 = 45 deg,
pi/6 = 30 deg.

(and things related to them, like 120, 135, etc.)

Write 11pi/12 = 165 deg = 120 deg + 45.
(could also be as  135 + 30, maybe others.)

cos(165) = cos 120 cos 45 - sin 120 sin 45.

sin(165) = sin 120 cos 45 + cos 120 sin 45.

sin 120 = sin 60 = s3/2
cos 120 = -cos 60 = - 1/2

sin 45 = cos 45 = s2/2

s2 means sqrt(2), etc.

For example:
cos(165) = cos 120 cos 45 - sin 120 sin 45.

cos(165) = -1/2 s2/2 - s3/2 s2/2

              s2(1 + s3)
cos(165) = -  -------------
                  4

I will leave the rest to you.  It's a bit of work, but you should be able to handle it:

we have cos(165)
compute sin(165)
compute sin 2T
compute cos 2T

Put them all together.

Paul Klarreich

Expertise

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

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I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

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