Question 720143
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please help me solve the following trigonometric equation using Pythagorean identity formula:

1 - sin^2theta = 0.5
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        The solution in the post by  @Theo is incomplete,  and therefore incorrect.

        I came to provide a correct solution.


        I will look for solutions in the interval  [0°,360°),  which represents the unit circle.



<pre>
I will use the letter t to represent {{{theta}}}.


The Pythagorean identify is sin^2(t) + cos^2(t) = 1.

You can write it in the form

    cos^2(t) = 1 - sin^2(t).


You are given that  1 - sin^2(t) = 0.5.

Hence,

    cos^2(t) = 0.5.


Take the square root of both sides of this equation to get:

    cos(t) = +/- {{{sqrt(0.5)}}}.


If  cos(t) = + {{{sqrt(0.5)}}},  then angle  't'  is either   45°  or -45°,  which geometrically is the same as 315°.

If  cos(t) = - {{{sqrt(0.5)}}},  then angle  't'  is either  135°  or  225°.



Thus, in the interval  [0°,360°)  the given equation  {{{1-sin^2(theta)}}} = 0.5  has 4 (four) solutions
      that are  45°,  135°,  225°  and  315°.
</pre>

Solved.


If a student losing the roots while solving such equation, &nbsp;as tutor @Theo did,
it is considered as a grave sin. &nbsp;The score is cut, &nbsp;and the student is sent for re-training.


And such teaching as presented in the post by &nbsp;@Theo is considered as unsatisfactory.