# URGENT - Trigonometry - Identities and Proofs

Okay, today, I find myself utterly dumbfounded by these three questions -
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!!!

1. 👍 0
2. 👎 0
3. 👁 141
1. In 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?

1. 👍 0
2. 👎 0
posted by Reiny
2. In 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

1. 👍 0
2. 👎 0
posted by Reiny

## Similar Questions

1. ### Geometry

Write a two-column proof. If point g is the centriod of triangle ABC and BG = 18. Find GE and BE. My work: ge is half so its 9. be equals 18 + 9 = 27 i need help on two column proof part

asked by Anthony on November 27, 2015
2. ### english

when and where did Euclid write the proof of the pythagorean theorem. why did the proof was written.

asked by 000 on April 2, 2011
3. ### Math

Please help me with this problem: A bourbon that is 51 proof is 25.5% alcohol by volume while one that is 82 proof is 41% alcohol. How many liters of 51 proof bourbon must be mixed with 1.0 L of 82 proof bourbon to produce a 66

asked by Lorie on April 5, 2013
4. ### Geometry

Write a two-column proof. Given: 7y=8x-14; y=6 Prove: x=7 I need help making a two-column proof with the statements and the reasons, please! Thank you! ( This is what I have so far) Statements | Reasons 1. 7y=8x-14;y=6 | Given 2.

asked by Unknown on September 7, 2016
5. ### college math

I posted this problem earlier, but I still can't figure it out. It is for extra credit. I was hoping if someone could please solve it. I would greatly appreciate it. A bourbon that is 51 proof is 25.5% alcohol by vol. while one

asked by Lorie on April 5, 2013
6. ### math

A bourbon that is 51-proof is 25.5% alcohol by volume while one that is 82-proof is 41% alcohol by volume. How many liters of 51 proof bourbon must be mixed with 1.0-L of 82 proof bourbon to produce a 66 proof bourbon? Give answer

asked by Lorie on April 4, 2013
7. ### English

Word box: alias amble burly distort dogged dumbfounded extinct fossil grit inevitable ingrained meteoric parody prevail rend replenish rummage skimp sleuth vandalism 2.We were nothing less than ____ when we saw the immense damage

asked by Jerald on February 26, 2013
8. ### communication

In general, thesis or central idea is supported by claims; claims are supported by proof; and A.proof is supported by development,including details and examples B.proof is supported by the specific purpose C.proof is supported by

asked by Michelle on November 13, 2011
9. ### Writing

I need help with writing an essay about simple questions, complicated questions, and complex questions. I need to make a topic, write 3-5 paragraphs, describe and explain, write it as an expository essay, and write one paragraph

asked by Jessica on October 28, 2014
10. ### Social Studies 9

Explain how television and other media can contribute to peoples hysteria. PLEASE HELP ME . ANOTHER question i don't understand is : Find out more about the conflict in the northern ireland today. write one account of the

asked by Stephanie on September 24, 2011

More Similar Questions