SOLUTION: find the exact value, if defined: cos((arcsec(-2)+arctan1)

Question 678945: find the exact value, if defined: cos((arcsec(-2)+arctan1)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
cos(arccsc(-2) + arctan(1))
Since the inverse functions return angles, this expression is the cos of the sum of two angles. We can use the cos(A+B) formula:
COS(A+B) = cos(A)cos(B) - sin(A)sin(B)
With the "A" being arccsc(-2) and the "B" being arctan(1) we get:
cos(arccsc(-2))cos(arctan(1)) - sin(arccsc(-2))sin(arctan(1))

arccsc(-2) is an angle whose csc is -2. Since sin and csc are reciprocals of each other, an angle whose csc is -2 will have a sin that is -1/2. If we didn't recognize that arccsc(-2) was a special angle, we should definitely recognize that an angle whose sin is -1/2 is a special angle. The reference angle will be 30 degrees (or radians). Arccsc only returns angles between -90 and 90. The only angle in this range which would have a sin of -1/2 is -30. So now our expression is:
cos(-30)cos(arctan(1)) - sin(-30)sin(arctan(1))

arctan(1) is an angle whose tan is 1. We should recognize that 1 is a special angle value for tan. We should know that the reference angle that has a tan of 1 is 45 degrees (or radians). Arctan, like arccsc, returns angles between -90 and 90. The only angle in this range that has a tan of positive 1 is 45. So now our expression is:
cos(-30)cos(45) - sin(-30)sin(45)

Now our expression consists of cos's and sin's of special angles. we should now what these are:

which simplifies to:

This is an exact expression for cos(arccsc(-2) + arctan(1)). It may be acceptable as your answer. Or, since the fractions have the same denominators, we could add them:

RELATED QUESTIONS
Find the exact value of 1.)y=tan inverse 1.) 2)y=cos inverse (-1) 3.)y=tan inverse... (answered by lwsshak3)

find the exact value, if defined:... (answered by lwsshak3)

Please help me with this problem Find the exact value of the expression, if it is... (answered by jsmallt9)

Find exact value of tan(arcsec 17�15 - cot... (answered by lwsshak3)

Find the exact value of the expression whenever it is defined. (If an answer is... (answered by lwsshak3)

Find the exact value of the expression, if it is defined. (If it is not defined, enter... (answered by stanbon)

if cos theta = -2/3 and 450 degrees < theta < 540degrees, find a. the exact value of... (answered by lwsshak3)

find the exact value, if defined. if cos2x=.8 and 2x is in QIV, find cosx. (answered by lwsshak3)

Find the exact value of cos(2arctan(-2)). (answered by lwsshak3)