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For the following problems, find the exact values of the six trigonometric functions: sine, cosine, tangent, cotangent, secent, & cosecent.  Please show your work on how you got to your answer.

 

  1. \((\sqrt(3)/2, 1/2)\)
  2. \((-2/5, \sqrt(21)/5)\)
  3. \((2\sqrt(2)/3,-1/3)\)
  4. \(\pi/4\)
  5. \(2\pi/3\)
  6. \(180°\)
 Dec 18, 2015
edited by gibsonj338  Dec 18, 2015

Best Answer 

 #1
avatar+128079 
+10

1.   x = sqrt (3) / 2    y = 1/2    r = 1     

 

So

 

sine  = y / r   = 1/2                       csc = 2 

cosine = x / r  =  [ sqrt(3) / 2]       sec = 2 / sqrt(3)

tangent = y/x =  sqrt(3)      cotangent = x /y  =  1/ sqrt(3)  =  sqrt(3) / 3

 

2. x = -2/5     y = sqrt(21) / 5    r =  1

 

sine  = sqrt(21)/5      csc = 5/sqrt(21)  = [5/21]*sqrt(21)

cosine  = -2/5            sec= -5/2

tangent  = -sqrt(21)/2     cotangent  = -2 / sqrt(21)   = [-2/21] * sqrt(21)

 

3.  x = [2sqrt(2)] /3      y =   - 1/3     r = 1

 

sine = -1/3      csc = -3

cosine  = [2sqrt(2)] /3    sec  = 3 / [2 sqrt(2)]  = [ 3 sqrt(2)] / 4

tangent = [-1] / [2 sqrt(2)]  = [- sqrt(2)] / [ 4]    cotangent  = -2sqrt(2)

 

4. 

 

sine pi/4   = sqrt2)/2      csc = sqrt(2)

cosine pi/4  =  same as sine     sec =  same as csc

tangent, cotangent pi/4   =  1cool

 

5.

 

sine 2pi/3   =   sqrt(3)/2      csc 2pi/3  =  2sqrt(3) / 3

cosine 2pi/3  = -1/2    sec2pi/3  = -2

tangent 2pi/3  = -sqrt(3)     cotangent 2pi/3 = -sqrt(3)/ 3

 

6.  

 

sine180°  = 0      csc 180°  = undefined

cosine 180° , sec 180°   = 1

tangent 180°  = 0      cotangent 180°  = undefined

 

 

 

cool cool

 Dec 18, 2015
 #1
avatar+128079 
+10
Best Answer

1.   x = sqrt (3) / 2    y = 1/2    r = 1     

 

So

 

sine  = y / r   = 1/2                       csc = 2 

cosine = x / r  =  [ sqrt(3) / 2]       sec = 2 / sqrt(3)

tangent = y/x =  sqrt(3)      cotangent = x /y  =  1/ sqrt(3)  =  sqrt(3) / 3

 

2. x = -2/5     y = sqrt(21) / 5    r =  1

 

sine  = sqrt(21)/5      csc = 5/sqrt(21)  = [5/21]*sqrt(21)

cosine  = -2/5            sec= -5/2

tangent  = -sqrt(21)/2     cotangent  = -2 / sqrt(21)   = [-2/21] * sqrt(21)

 

3.  x = [2sqrt(2)] /3      y =   - 1/3     r = 1

 

sine = -1/3      csc = -3

cosine  = [2sqrt(2)] /3    sec  = 3 / [2 sqrt(2)]  = [ 3 sqrt(2)] / 4

tangent = [-1] / [2 sqrt(2)]  = [- sqrt(2)] / [ 4]    cotangent  = -2sqrt(2)

 

4. 

 

sine pi/4   = sqrt2)/2      csc = sqrt(2)

cosine pi/4  =  same as sine     sec =  same as csc

tangent, cotangent pi/4   =  1cool

 

5.

 

sine 2pi/3   =   sqrt(3)/2      csc 2pi/3  =  2sqrt(3) / 3

cosine 2pi/3  = -1/2    sec2pi/3  = -2

tangent 2pi/3  = -sqrt(3)     cotangent 2pi/3 = -sqrt(3)/ 3

 

6.  

 

sine180°  = 0      csc 180°  = undefined

cosine 180° , sec 180°   = 1

tangent 180°  = 0      cotangent 180°  = undefined

 

 

 

cool cool

CPhill Dec 18, 2015

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