precal help

If the cos of an angle is positive, the angle will be in either the first quadrant or the fourth quadrant.

If the csc of an angle is negative (since csc = 1/sin), the sin will also be negative and the sin is negative in either the third quadrant or the fourth quadrant.

 

Since cos(t) = 4/7 and csc(t) < 0, tthe angle t must be in the fourth quadrant.

 

Since cos = x/r and cos(t) = 4/7,  x = 4 and r = 7.

Using the Pythagorean Theorem:  x² + y²  =  r²    --->   4² + y²  =  7²​   --->   16 + y²  =  49   --->   y²  =  33

So, either  y = sqrt(33)  or  y  =  - sqrt(33).

Since the angle is in the fourth quadrant:  y  =  -sqrt(33).

 

sin = y/r   --->   sin(t)  =  -sqrt(33) / 7

tan = y/x   --->   tan(t)  =  -sqrt(33) / 4