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July 5, 2015

July 5, 2015

Posted by **Veronica** on Saturday, April 2, 2011 at 12:01am.

Write a proof for -

2/(sqrt(3)cos(x) + sin(x))= sec((pi/6)-x)

Solve the following equation -

2sin(2x) - 2sin(x) + 2(sqrt(3)cos(x)) - sqrt(3) = 0

Find all solutions (exact) to the equation -

sin^2(x)cos^2(x) = (2 - sqrt(2))/16

Any Help at all will most definitely be appreciated!!!

- URGENT - Trigonometry - Identities and Proofs -
**Reiny**, Saturday, April 2, 2011 at 8:59amIn proving identities the general rule is to start with the more complicated side and, by mathematical manipulation, show that it is equal to the other side.

(even though it may not look it, in your problem the rights side (RS) is more complicated, so ....

RS = 1/cos(π/6-x)

= 1/cos(30°-x) , it makes no difference if you work in degrees or radians

= 1/(coss30cosx + sin30sinx)

= 1/((√3/2)cosx + (1/2)sinx)

= 1/((√3cosx + sinx)/2)

= 2/(√3cosx + sinx)

= LS

Do either solve or prove these type of equations, you have to be familiar with the basic trig relationships, such as

cos(A±B) = cosAcosB -/+ sinAsinB, which I used above.

I also needed to know the trig ratios of the standard 30-60-90 right-angled triangle.

in the first of your "solve equation" we will need

sin(2A) = 2sinAcosA

2sin(2x) - 2sinx + 2√3cosx - √3 = 0

4sinxcosx - 2sinx +2√3cosx - √3 = 0

common factors

2sinx(2cosx - 1) + √3(2cosx - 1) = 0

(2cosx - 1)(2sinx + √3) = 0

2cosx-1 = 0 or 2sinx + √3 = 0

cosx = 1/2 or sinx = -√3/2

the last of these is not possible since the sine and the cosine fall between -1 and +1

so cosx = 1/2

from the 30-60-90 triangle , we know cos 60° = 1/2

and by the CAST rule, the cosine is positive in I and IV

so x = 60° or x = 300°

or

x = π/3 or x = 5π/3 radians.

last one:

sin^2x cos^2x = (2-√2)/16

take √ of both sides

sinxcosx = ±√(2-√2)/4

4sinxcosx = ±√(2-√2

2sin 2x = ±√(2-√2)

sin2x = ±√(2-√2)/2

so 2x is in any of the 4 quadrants

using my calculator , I evaluated the right side, then by "inverse sin", found

2x = 22.5° ( ahh, 1/2 of the 45, one of the standard angles)

so 2x = 22.5 or 157.5 or 202.5 or 337.5

making

x = 11.25 , 78.75 , 101.25 or 168.75

since the period of sin2x , which gave us our answer, is 180/2 or 90°, adding or subtracting multiples of 90 to any of the above answers will produce a new answer.

e.g. 78.75+90= 168.75° will also be an answer.

let's test that:

sin^2 (168.75) + cos^2(168.75)

= .036611651 by calculator

RS = (2-√2)/16 = .036611652 , not bad eh?

- Appendex - URGENT - Trigonometry - Identities and Proofs -
**Reiny**, Saturday, April 2, 2011 at 9:03amIn the second equation I rejected

sinx = -√3/2

I shouln't have since that value lies between -1 and 1

from the above we also get

x = 120° or x =240°

or

x = 2π/3 or x = 4π/3 rads