Question 325522
Find all solutions for the following equations.Record your answer in exact form where possible.
(a) {{{2}}}{{{"cos"^2x}}}{{{""=""}}}{{{-3}}}{{{"sin"}}}{{{x}}}
(b) {{{"sin"}}}{{{x}}}{{{""*""}}}{{{"tan"}}}{{{x}}}{{{""=""}}}{{{"sin"}}}{{{x}}}
 
 Thinking I may have to us Half-angle formula on these (not sure)
<pre><b>
No you don't need those.

{{{2}}}{{{"cos"^2x}}}{{{""=""}}}{{{-3}}}{{{"sin"}}}{{{x}}}

Use the identity:  {{{"sin"^2theta}}}{{{""+""}}}{{{"cos"^2theta}}}{{{""=""}}}{{{1}}} written as {{{"cos"^2theta}}}{{{""=""}}}{{{1}}}{{{""-""}}}{{{"sin"^2theta}}} to rewrite the left side:

{{{2}}}{{{(1-"sin"^2x)}}}{{{""=""}}}{{{-3}}}{{{"sin"}}}{{{x}}}

{{{2}}}{{{""-""}}}{{{2}}}{{{"sin"^2x}}}{{{""=""}}}{{{-3}}}{{{"sin"}}}{{{x}}}

Get 0 on the right:

{{{2}}}{{{""-""}}}{{{2}}}{{{"sin"^2x}}}{{{""+""}}}{{{3}}}{{{"sin"}}}{{{x}}}{{{""=""}}}{{{0}}}

Rearrange terms on the left in descending order:

{{{-2}}}{{{"sin"^2x}}}{{{""+""}}}{{{3}}}{{{"sin"}}}{{{x}}}{{{""+""}}}{{{2}}}{{{""=""}}}{{{0}}}

To make factoring easier, multiply through by -1

{{{2}}}{{{"sin"^2x}}}{{{""-""}}}{{{3}}}{{{"sin"}}}{{{x}}}{{{""-""}}}{{{2}}}{{{""=""}}}{{{0}}}

{{{"(sin"}}}{{{x}}}{{{""-""}}}{{{"2)("}}}{{{2}}}{{{"sin"}}}{{{x}}}{{{""+""}}}{{{"1)"}}}{{{""=""}}}{{{0}}}

Setting the first factor = 0

{{{"sin"}}}{{{x}}}{{{""-""}}}{{{2}}}{{{""=""}}}{{{0}}}

{{{"sin"}}}{{{x}}}{{{""=""}}}{{{2}}}

This is impossible because no sine can be greater than 1 or
less than -1.

Setting the second factor = 0

{{{2}}}{{{"sin"}}}{{{x}}}{{{""+""}}}{{{"1"}}}{{{""=""}}}{{{0}}}

{{{2}}}{{{"sin"}}}{{{x}}}{{{""=""}}}{{{"-1"}}}

{{{"sin"}}}{{{x}}}{{{""=""}}}{{{-1/2}}}

All angles in QIII and QIV  with reference angle 30° (or {{{pi/6}}}
if your teacher wants radians) are solutions.

In degrees:
{{{x}}}{{{""=""}}}{{{"210°+n*360°"}}}; {{{x}}}{{{""=""}}}{{{"330°+n*360°"}}}

In radians:
{{{x}}}{{{""=""}}}{{{7pi/6+2pi*n}}};{{{x}}}{{{""=""}}}{{{11pi/6+2pi*n}}}
 
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{{{"sin"}}}{{{x}}}{{{""*""}}}{{{"tan"}}}{{{x}}}{{{""=""}}}{{{"sin"}}}{{{x}}}

Get 0 on the right:

{{{"sin"}}}{{{x}}}{{{""*""}}}{{{"tan"}}}{{{x}}}{{{""-""}}}{{{"sin"}}}{{{x}}}{{{""=""}}}{{{0}}}

Factor out {{{"sin"}}}{{{x}}} on the left side:

{{{"sin"}}}{{{x}}}{{{"(tan"}}}{{{x}}}{{{""-""}}}{{{"1)"}}}{{{""=""}}}{{{0}}}

Setting the first factor = 0

{{{"sin"}}}{{{x}}}{{{""=""}}}{{{0}}}

All angles with reference angle 0 are solutions:

In degrees:
{{{x}}}{{{""=""}}}{{{"n*180°"}}}

In radians:
{{{x}}}{{{""=""}}}{{{n*pi}}};

Setting the second factor = 0

{{{"tan"}}}{{{x}}}{{{""-""}}}{{{"1"}}}{{{""=""}}}{{{0}}}

{{{"tan"}}}{{{x}}}{{{""=""}}}{{{"1"}}}

All angles in QI and QIII  with reference angle 45° (or {{{pi/4}}}
if your teacher wants radians) are solutions.

In degrees:
{{{x}}}{{{""=""}}}{{{"45°+n*360°"}}}; {{{x}}}{{{""=""}}}{{{"225°+n*360°"}}}

In radians:
{{{x}}}{{{""=""}}}{{{pi/4+2pi*n}}};{{{x}}}{{{""=""}}}{{{7pi/4+2pi*n}}}

Edwin</pre>