If sin theta = 2/7 and theta is in quadrant 2 find cos theta, sec theta, csc theta, tan theta, and cot theta
sin(Θ) = y/r
x = √(r^2 - y^2) = √(49 - 4) = √45 = 3√5
cos(Θ) = x/r ... sec(Θ) = r/x ... csc(Θ) = r/y ... tan(Θ) = y/x ... cot(Θ) = x/y
careful with the sign.
x^2 + y^2 = r^2
x^2 + 4 = 49
x^2 = 45
x = ± 3√5
but since the angle is in quadrant II,
x = -3√5, y = 2 , r = 7
now sub into the formulas given by R_scott
To find the values of cos theta, sec theta, csc theta, tan theta, and cot theta, we need to use the information given: sin theta = 2/7 and theta is in quadrant 2.
In quadrant 2, the cosine function (cos theta) is negative. So, we cannot directly use the given value of sin theta to find cos theta. Instead, we will use the Pythagorean identity to find the missing value.
The Pythagorean identity is: sin^2(theta) + cos^2(theta) = 1.
Substituting sin theta = 2/7 into the equation, we have:
(2/7)^2 + cos^2(theta) = 1
4/49 + cos^2(theta) = 1
cos^2(theta) = 45/49
cos(theta) = ± √(45/49)
Since theta is in quadrant 2, where cosine is negative, we take the negative value:
cos(theta) = -√(45/49) = -√45/7
Now, let's find the other trigonometric functions:
sec theta is the reciprocal of cos theta:
sec(theta) = 1/cos(theta) = -1/(√45/7) = -7/√45 = -7√45/45
csc theta is the reciprocal of sin theta:
csc(theta) = 1/sin(theta) = 1/(2/7) = 7/2
tan theta is the ratio of sin theta to cos theta:
tan(theta) = sin(theta)/cos(theta) = (2/7)/(-√45/7) = -2/√45 = -2√45/45
cot theta is the reciprocal of tan theta:
cot(theta) = 1/tan(theta) = 1/(-2√45/45) = -45/2√45 = -45/(2√5)
Therefore, in quadrant 2, when sin theta = 2/7:
cos theta = -√45/7
sec theta = -7√45/45
csc theta = 7/2
tan theta = -2√45/45
cot theta = -45/(2√5)