Unit Circle Problem: Finding Six Trig Functions?

the point (-1, 5) is located in quadrant 2. to calculate all the trig functions, we're going to need the hypotenuse of the triangle the point forms with the x-axis and the origin, so let's calculate that first. use the pythagorean theorem.

sqrt((-1)^2 + (5)^2) = sqrt(1 + 25) = sqrt(26)

to calculate the trig functions, we can use theta as the angle between the negative x-axis and the line connecting (0,0) to (-1,5). that means the side opposite the angle has length 5, the side adjacent to the angle corresponds to the -1, and the hypotenuse is sqrt(26)

sin(theta) = opposite/hypotenuse = 5/sqrt(26)

cos(theta) = adjacent/hypotenuse = -1/sqrt(26)

tan(theta) = opposite/adjacent = 5/-1 = -5

csc(theta) = hypotenuse/opposite = sqrt(26)/5

sec(theta) = hypotenuse/adjacent = sqrt(26)/-1 = -sqrt(26)

cot(theta) = adjacent/opposite = -1/5

your teacher may want you to get the square root out of the denominator, which means

sin(theta) = 5/sqrt(26) = 5sqrt(26)/26

cos(theta) = -1/sqrt(26) = -sqrt(26)/26

the element (a million,-a million) places us interior the 4th quadrant: x = a million y = -a million r = sqrt[x^2+y^2] = sqrt[a million^2 + (-a million)^2] = sqrt[a million+a million] = sqrt2 r = sqrt[2] sin(theta) = y/r sin(theta) = -a million/sqrt2 = -sqrt2/2 cos(theta) = x/r cos(theta) = a million/sqrt2 = sqrt2/2 tan(theta) = y/x tan(theta) = -a million/a million = -a million csc(theta) = a million/sin(thetha) = r/y csc(theta) = sqrt2/-a million = -sqrt2 sec(theta) = a million/cos(theta) = r/x sec(theta) = sqrt2/a million = sqrt2 cot(theta) = a million/tan(theta) = x/y cot(theta) = a million/-a million = -a million voila