If sin theta is equal to 5/13 and theta is an angle in quadrant II find the value of cos theta, sec theta, tan theta, csc theta, cot theta.
make a sketch of your triangle
since sinØ = opposite /hypotenuse = 5/13
opposite = 5 and hypotenuse = 13
by Pythagoras:
x^2 + 5^2 = 13^2
x^2 +25 = 169
x^2 = 144
x = ± 12 , but in quadrant II, x = - 12
sinØ = 5/13 , cscØ = 13/5
cosØ = -12/13 , secØ = - 13/12
tanØ = 5/-12 = -5/12 , cot Ø = -12/5
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To find the values of cos theta, sec theta, tan theta, csc theta, and cot theta, we can use the given information that sin theta is equal to 5/13 and theta is in quadrant II.
Using the Pythagorean identity, we can determine the value of cos theta:
cos^2(theta) + sin^2(theta) = 1
cos^2(theta) + (5/13)^2 = 1
cos^2(theta) + 25/169 = 1
cos^2(theta) = 1 - 25/169
cos^2(theta) = 144/169
Taking the square root of both sides, we can find the value of cos theta:
cos(theta) = ± √(144/169)
Since theta is in quadrant II, where cosine values are negative, we take the negative square root:
cos(theta) = - 12/13
Using the reciprocal identities, we can find the values of sec theta, tan theta, csc theta, and cot theta:
sec(theta) = 1/cos(theta)
sec(theta) = 1/(-12/13)
sec(theta) = -13/12
tan(theta) = sin(theta)/cos(theta)
tan(theta) = (5/13)/(-12/13)
tan(theta) = -5/12
csc(theta) = 1/sin(theta)
csc(theta) = 1/(5/13)
csc(theta) = 13/5
cot(theta) = 1/tan(theta)
cot(theta) = 1/(-5/12)
cot(theta) = -12/5
So, the values of cos theta, sec theta, tan theta, csc theta, and cot theta are -12/13, -13/12, -5/12, 13/5, and -12/5 respectively.
To find the values of cos theta, sec theta, tan theta, csc theta, and cot theta, given that sin theta is equal to 5/13 and theta is in quadrant II, we can use the Pythagorean Identity and the definitions of trigonometric functions.
1. Start by using the Pythagorean Identity: sin^2(theta) + cos^2(theta) = 1. Since sin theta is 5/13, we can replace sin^2(theta) with (5/13)^2 in the equation:
(5/13)^2 + cos^2(theta) = 1
2. Solve for cos^2(theta):
cos^2(theta) = 1 - (5/13)^2
cos^2(theta) = 1 - 25/169
cos^2(theta) = 144/169
3. Take the square root of both sides to find cos theta:
cos(theta) = sqrt(144/169)
cos(theta) = 12/13
4. To find sec theta, you can use the reciprocal identity: sec(theta) = 1/cos(theta):
sec(theta) = 1 / (12/13)
sec(theta) = 13/12
5. To find tan theta, you can use the definition of tangent: tan(theta) = sin(theta) / cos(theta):
tan(theta) = (5/13) / (12/13)
tan(theta) = 5/12
6. To find csc theta, you can use the reciprocal identity: csc(theta) = 1/sin(theta):
csc(theta) = 1 / (5/13)
csc(theta) = 13/5
7. To find cot theta, you can use the reciprocal identity: cot(theta) = 1/tan(theta):
cot(theta) = 1 / (5/12)
cot(theta) = 12/5
So, the values of cos theta, sec theta, tan theta, csc theta, and cot theta when sin theta is equal to 5/13 and theta is in quadrant II are:
cos(theta) = 12/13
sec(theta) = 13/12
tan(theta) = 5/12
csc(theta) = 13/5
cot(theta) = 12/5