Given csc theta= 4, cot theta < 0
Find the exact values of cos theta and tan theta
since sin>0 and cos<0, we are in QII.
So, y=1,r=4,x=-√15
sin = y/r = 1/4
cos = x/r = -√15/4
tan = y/x = -1/√15
...
please help me
csc(theta)=4 and cot(theta)<0
To find the exact values of cos(theta) and tan(theta), we can use the given information about csc(theta) and cot(theta).
First, let's find the value of sin(theta) using the given value of csc(theta). We know that csc(theta) is the reciprocal of sin(theta), so we have:
csc(theta) = 1/sin(theta) = 4
To find sin(theta), we can take the reciprocal of csc(theta):
sin(theta) = 1/csc(theta) = 1/4
Next, we can find the value of cos(theta) using the Pythagorean identity:
cos^2(theta) = 1 - sin^2(theta)
Plugging in the value of sin(theta) we found earlier, we have:
cos^2(theta) = 1 - (1/4)^2 = 1 - 1/16 = 15/16
Taking the square root of both sides gives us:
cos(theta) = +/- √(15/16)
Since cos(theta) can be positive or negative, we need to determine its sign. Since csc(theta) = 4 is positive but cot(theta) < 0, we know that theta is in the second quadrant where both sin(theta) and cos(theta) are positive. Therefore, we can take the positive square root:
cos(theta) = √(15/16) = √15/4
Finally, we can find the value of tan(theta) using the given information about cot(theta). We know that cot(theta) is the reciprocal of tan(theta), so we have:
cot(theta) = 1/tan(theta) < 0
Since cot(theta) is negative, we know that tan(theta) is also negative. Thus:
tan(theta) = -1
Therefore, the exact values of cos(theta) and tan(theta) are:
cos(theta) = √15/4
tan(theta) = -1