Find the exact value of the following expression.

Question

sin^-1 (-√2)

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Answer 1

If you are dealing with the inverse sine FUNCTION, the range is [-π/2, π/2].  So, Sin^-1(-√2/2) is the angle in the interval

[-π/2, π/2] whose sine is -√2/2.

 

So, Sin^-1(-√2/2) = -π/4.  This is the value (in approximate decimal form) returned by a calculator if Sin^-1(-√2/2) is entered and the calculator is in radian mode.  In degree mode, -45° would be the calculator's output.  

Answer 2

Hi Ky,

 

Recall sin^-1 is the inverse sin function.  The question is really asking you what angle has a sin of (-√2)/2.  If you are familiar with the unit circle you know that sin 45 degrees (or π/4) is (√2)/2.  But we need to find the angle when the sin is negative (-√2)/2.  Sin is negative in the 3rd and 4th quadrants.  With 45 degrees or π/4 as a reference angle the answer is 5π/4 (225 degrees) or 7π/4 (315 degrees).

Answer 3

First, determine which angle has a value of √2/2 for the sine function. From the 45-45-90 triangle, we know that sin Θ = 1/√2 = √2/2.

 

Now, ask yourself in which quadrants is the sine function negative. The sine function is negative in Quadrants III and IV.

 

You should now be able to use these two pieces of information to determine the exact value of the expression. Your answer, of course, will depend upon whether Θ is restricted over a specified interval.