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what values for Θ (0 < Θ < 2 pi) satisfy the equation? cos Θ- tan Θ = 0

Guest Apr 25, 2015

Best Answer 

 #2
avatar+26750 
+5

You could also solve this algebraically as follows.  Knowing that tan θ = sinθ/cosθ we can write:

 

cosθ - sinθ/cosθ = 0

 

Multiply through by cosθ

cos2θ - sinθ = 0

 

Also, cos2θ = 1 - sin2θ so

1 - sin2θ - sinθ = 0

 

Rearrange:

sin2θ + sinθ - 1 = 0

 

This is a quadratic in sinθ with solutions -(1-√5)/2 and -(1+√5)/2.  The second result is not a valid solution because it is outside the range ±1.

 

So taking the arcsine of the other result we get

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\mathtt{\,-\,}}{\frac{\left({\mathtt{1}}{\mathtt{\,-\,}}{\sqrt{{\mathtt{5}}}}\right)}{{\mathtt{2}}}}\right)} = {\mathtt{38.172\: \!707\: \!627\: \!012^{\circ}}}$$

 

There is another value at 180°-38.173° = 141.827°

.

Alan  Apr 26, 2015
 #1
avatar+87301 
+5

cosΘ - tanΘ   =0     add tanΘ  to both sides

cosΘ  = tanΘ

The easiest way to solve this - IMHO - is with a graph.......here it is....

https://www.desmos.com/calculator/wvqydtk4mt

The points of intersection occur at about 38.2°  and at about 141.8°

And converting to rads we have  ....about .6667 rads and about 2.475 rads on  (0, 2pi)

 

  

CPhill  Apr 25, 2015
 #2
avatar+26750 
+5
Best Answer

You could also solve this algebraically as follows.  Knowing that tan θ = sinθ/cosθ we can write:

 

cosθ - sinθ/cosθ = 0

 

Multiply through by cosθ

cos2θ - sinθ = 0

 

Also, cos2θ = 1 - sin2θ so

1 - sin2θ - sinθ = 0

 

Rearrange:

sin2θ + sinθ - 1 = 0

 

This is a quadratic in sinθ with solutions -(1-√5)/2 and -(1+√5)/2.  The second result is not a valid solution because it is outside the range ±1.

 

So taking the arcsine of the other result we get

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\mathtt{\,-\,}}{\frac{\left({\mathtt{1}}{\mathtt{\,-\,}}{\sqrt{{\mathtt{5}}}}\right)}{{\mathtt{2}}}}\right)} = {\mathtt{38.172\: \!707\: \!627\: \!012^{\circ}}}$$

 

There is another value at 180°-38.173° = 141.827°

.

Alan  Apr 26, 2015

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