We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
52
5
avatar

If \(0 < \theta < \frac{\pi}{2}\) and \(\sqrt{3} \cos \theta - \sin \theta = \frac{1}{3},\) then find \(\sqrt{3} \sin \theta + \cos \theta.\)

 Nov 5, 2019
 #1
avatar+105411 
+2

√3 cos θ  - sin θ  =  1/3

 

 

 √3 cos θ  =  sin θ  +1/3       square both sides

 

3cos^2θ  =  sin^2θ + (2/3)sin θ + 1/9

 

3 (1 - sin^2θ)  = sin^2 θ + (2/3)sin θ  + 1/9 

 

3 - 3sin^2 θ = sin^2θ + (2/3)sin θ + 1/9         rearrange as

 

4sin^2 θ + (2/3) sinθ - 26/9  =  0             multiply through by 9

 

36sin^2θ +6sin θ - 26  =   0     

 

18 sin^2 θ + 3sin θ - 13  = 0          let   sin θ  =  x

 

18x^2  + 3x  -  13  =  0

 

Using the quadratic formula  and taking the poisitve result  [ since the sine is positive here, we get that ]

 

x  = sin θ  =  - 3   + √[ 3^2 + 4* 18 * 13]

                    ______________________  ≈  .77

                              2 * 18

 

 

Using the sine inverse  we get that 

 

arcsin (.77)  = θ  ≈  .88 rads

 

So

 

3 sin (.88)  + cos (.88)  ≈  1.972

 

 

 

cool cool cool

 Nov 5, 2019
edited by CPhill  Nov 5, 2019
edited by CPhill  Nov 6, 2019
 #2
avatar+2499 
+1

I do not understand any of those trig gibberish yet, but nice job Cphill! One day I will learn the dark secrets of sine cosine and tangent!

CalculatorUser  Nov 6, 2019
edited by CalculatorUser  Nov 6, 2019
 #3
avatar+105411 
0

Don't worry, CU......as good as you are in math, the trig "secrets"  will come easily to you....!!!!

 

 

 

cool cool cool

CPhill  Nov 6, 2019
 #4
avatar
+1

sorry cphill but its \(\frac{\sqrt{35}}{3}\)

Guest Nov 6, 2019
 #5
avatar+105411 
+1

I just rounded

 

The quadratic  evaluates to

 

[√105 - 1 ]

________    =    sin  θ

     12

 

 

So

 

√3sin  ( arcsin ( [√105 - 1 ]/12 ) )  +  cos (arcsin ( [√105 - 1 ] /12) )  =

 

√35

 ___   ≈   1.972

   3

 

 

cool cool cool

CPhill  Nov 6, 2019

7 Online Users

avatar