 Hello and welcome to the session. In this session, we will discuss a question which says that find sin of cos inverse of minus 0.5. Now before starting the solution of this question, we should know a result and here we will see signs of the trigonometric functions on coding appearing. Now in the first quadrant, all functions, that is all trigonometric functions are positive. In the second quadrant, sin functions are positive. In the third quadrant, tangent, cotangent, secant, this means only sin and cosecant are positive in the second quadrant. Rest all trigonometric functions are negative. In the third quadrant, tangent and cotangent are positive and rest all the trigonometric functions are negative. And in the fourth quadrant, cosine and secant are positive and rest all of the functions are negative. But in the first quadrant, all the trigonometric functions are positive. Now this result will work out as a key idea for solving out the given question. Now let us start with the solution of the given question. Here we have to find sin inverse of minus 0.5. Now let theta is equal to cosecant inverse of minus 0.5 by definition of inverse trigonometric functions. We have theta is equal to minus 0.5 cos theta is equal to minus 5 upon 10 which is equal to minus 1 by 2. Now here, cos theta is negative. Now first of all, let us find reference angle that is a cubed angle in first quadrant where cos theta is equal to 1 by 2 as we know from the key idea that all trigonometric functions are positive in the first quadrant. So here we will find reference angle in the first quadrant where cos theta is equal to 1 by 2 which is positive. These angle are in radiance pi by 3 cos is positive. Now cos 60 degrees is equal to 1 by 2 in second quadrant at an angle of 180 degrees which is equal to 20 degrees is negative as we know that it is negative in the second quadrant 20 degrees is equal to cos of 180 degrees minus 60 which is equal to minus 1 by 2. Now we know that 120 degrees is equal to 180 degrees minus 60 degrees. Now in radiance we can write it as pi minus 2 pi by 3 and this is equal to 2 pi by 3. So 1 degree angle in radiance 2 pi by 3. Now here we are writing cos of 120 degrees as minus 1 by 2. This means cos d is equal to minus 1 by 2 and theta is equal to 2 pi by 3. Now since we have cos of minus 0.5 equal to cos inverse of minus 0.5 therefore minus 0.5 of minus 0.5 2 pi by 3 and from the key idea we know that is positive in second quadrant sin is positive. So this is equal to in the table of values of trigonometric functions we know that sin pi by 3 is square root of 3 by 2 inverse of minus 0.5 is equal to square root of 3 of write the session.