Question 825934
When a problem says "find the exact value" or "do not use a calculator" it usually means that the problem involves special angles.<br>
Here's a solution:<ul><li>With {{{sin^(-1)(-sqrt(3)/2)}}} we should recognize (even without being told not to use a calculator) that {{{sqrt(3)/2}}} is a special angle value for sin.</li><li>We should also know that a reference angle of {{{pi/3}}} has a sin of {{{sqrt(3)/2}}}.</li><li>Since the angle in question, {{{sin^(-1)(-sqrt(3)/2)}}}, has a negative sin value and since sin is negative in the 3rd and 4th quadrants, we now know that {{{sin^(-1)(-sqrt(3)/2)}}} terminates in one of these quadrants.</li><li>We should know that the inverse sin function has a range limited to angles between {{{-pi/2}}} and {{{pi/2}}}. So {{{sin^(-1)(-sqrt(3)/2)}}} will have a value in that range. (Note: This range includes angles which terminate in the 1st quadrant (between 0 and {{{pi/2}}}) and the 4th quadrant (between 0 and {{{-pi/2}}}).</li><li>Combining the fact that the angle must terminate in the 3rd or 4th quadrants (because the sin is negative) and the range of the inverse sin function, we should now know that {{{sin^(-1)(-sqrt(3)/2)}}} is between 0 and {{{-pi/2}}}</li><li>Combining that with the fact that the reference angle is {{{pi/3}}}, it should not take long to figure out that:
{{{-pi/3}}}
is the <u>only</u> angle with the right reference angle for {{{sqrt(3)/2}}}, with a negative sin and in the range of the inverse sin function.</li></ul>To turn this into degrees, just multiply it by {{{180/pi}}}.