Can someone help me find the exact value of 4csc(3pi/4)-cot(-pi/4)?
Thanks!
cotx =1/tanx
cscx = 1/sinx
If you find it easier to conceptualize in degrees, realize that pi/4 radians is 45º and 3pi/4 is then 135º
If you know the CAST rule, it is easy to see that -pi/4 is in the fourth quadrant and tan(-pi/4) or tan(-45º) = -1
therefore cot(-pi/4) = -1
3pi/4 or 135º is in the second quadrant where the sine is positive.
so sin(3pi/4) = 1/(√2)
then csc(3pi/4) = √2
and 4csc(3pi/4) = 4√2
so the exact value of 3csc(3pi/4) - cot(-pi/4)
= 4√2 + 1
To find the exact value of 4csc(3pi/4) - cot(-pi/4), you can use the definitions of cotangent (cot) and cosecant (csc) in terms of other trigonometric functions.
First, let's find the value of cot(-pi/4). The cotangent of an angle is equal to 1 divided by the tangent of that angle. In this case, -pi/4 represents an angle in the fourth quadrant, where the tangent is negative. Therefore, the tangent of -pi/4 is -1.
So, cot(-pi/4) = 1/tan(-pi/4) = 1/(-1) = -1.
Next, let's find the value of csc(3pi/4). The cosecant of an angle is equal to 1 divided by the sine of that angle. In this case, 3pi/4 represents an angle in the second quadrant, where the sine is positive. Therefore, the sine of 3pi/4 is positive. To find the exact value, we can use the fact that 3pi/4 is equivalent to 135 degrees. In degrees, the sine of 135 degrees is (1/√2).
So, csc(3pi/4) = 1/sin(3pi/4) = 1/(1/√2) = √2.
Now that we have the values of cot(-pi/4) = -1 and csc(3pi/4) = √2, we can substitute them into the equation:
4csc(3pi/4) - cot(-pi/4) = 4(√2) - (-1) = 4√2 + 1.
Therefore, the exact value of 4csc(3pi/4) - cot(-pi/4) is 4√2 + 1.