Let theta be an angle in quad III such that sec theta = -13/5. Find the exact values of cot theta and sin theta?
Thank you!!!
sec Ø = -13/5 in quad II
so cosØ = -5/13
x = -5, r = 13 and y = +12
sinØ = 12/13
cotØ = -5/12
To find the exact values of cot theta and sin theta, we first need to determine the cosine of theta.
We know that sec theta is equal to -13/5, and secant is the reciprocal of cosine. Hence, we can calculate cosine theta as the reciprocal of -13/5:
cos theta = 1 / (sec theta) = 1 / (-13/5) = -5/13
Now that we know the cosine of theta, we can find the sine of theta by using the Pythagorean identity:
sin^2 theta + cos^2 theta = 1
Substituting the value of cos theta we just found,
sin^2 theta + (-5/13)^2 = 1
sin^2 theta + 25/169 = 1
sin^2 theta = 1 - 25/169
sin^2 theta = 144/169
Taking the square root of both sides, we find:
sin theta = sqrt(144/169) = 12/13
Now, to find the cotangent of theta, we can use the expression:
cot theta = cos theta / sin theta
Substituting the values we found earlier,
cot theta = (-5/13) / (12/13)
cot theta = -5/12
So, the exact values of cot theta and sin theta are -5/12 and 12/13 respectively.
To find the exact values of cot(theta) and sin(theta), we can use the relationship between trigonometric functions and the definitions of secant and cosine.
Given that sec(theta) = -13/5, we know that sec(theta) = 1/cos(theta).
Using this information, we can find cos(theta) by taking the reciprocal of sec(theta):
cos(theta) = 1 / sec(theta)
= 1 / (-13/5)
= -5/13
Since theta is in quadrant III, where cosine is negative, we have cos(theta) = -5/13.
Now, we can use the Pythagorean identity to find sin(theta):
sin^2(theta) + cos^2(theta) = 1
Plugging in the value of cos(theta) we just found:
sin^2(theta) + (-5/13)^2 = 1
sin^2(theta) + 25/169 = 1
sin^2(theta) = 1 - 25/169
sin^2(theta) = 144/169
Taking the square root of both sides, and considering that theta is in quadrant III where sine is also negative:
sin(theta) = -√(144/169)
= -12/13
Finally, we can find the value of cot(theta) using the reciprocal relationship between cotangent and tangent:
cot(theta) = 1 / tan(theta)
Since tan(theta) = sin(theta) / cos(theta), we can substitute the values we found earlier:
cot(theta) = 1 / (sin(theta) / cos(theta))
= cos(theta) / sin(theta)
= (-5/13) / (-12/13)
= 5/12
Therefore, the exact values of cot(theta) and sin(theta) are cot(theta) = 5/12 and sin(theta) = -12/13, respectively.
If you have any further questions, feel free to ask!