SOLUTION: let theta be angle in quadrant II such that sec(theta)=-10/7. Find the exact values of cot(theta) and sin(theta)

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Question 855428: let theta be angle in quadrant II such that sec(theta)=-10/7. Find the exact values of cot(theta) and sin(theta)
Found 2 solutions by lwsshak3, Theo:
Answer by lwsshak3(11628) (Show Source):
You can put this solution on YOUR website!
let theta be angle in quadrant II such that sec(theta)=-10/7. Find the exact values of cot(theta) and sin(theta)
***
use x for theta
secx=-10/7
cosx=-7/10
sinx=√(1-cos^2x)=√(1-49/100)=√(51/100)=√51/10
cotx=cosx/sinx=-7/√51=-7√51/51
..
calculator check:
cosx=-7/10
x≈134.43˚
..
sinx=sin(134.43)≈0.7141
exact value=√51/10≈0.7141
..
cotx=cot(134.43)=-0.9803
exact value=-7√51/51≈0.9802

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
T = Theta.
sec(T) = -10/7
this means that the hypotenusse is 10 and the adjacent side of the angle is -7.
you can find the opposite side of the angle by the pythagorean formula.
c^2 = a^2 + b^2
c = 10
a = -7
from that formula, solve for b^2 to get:
b^2 = c^2 - a^2
substitute for c and a in that formula to get:
b^2 = (10)^2 - (-7)^2 to get:
b^2 = 100 - 49 which becomes:
b^2 = 51 which makes:
b = sqrt(51)

you now have:
angle = T
hypotenuse = 10
a = -7
b = sqrt(51)

you want to find cot(T):

cot(T) = adjacent / opposite = -7/sqrt(51)

you want to find sin(T):

sin(T) = opposite / hypotenuse = sqrt(51) / 10.

you can use your calculator to confirm that you did all this correctly.

my method is as follows:

assume the angle is in quadrant 1.
then all of the functions will be positive.
so solve for the angle as if the angle was in quadrant 1.
you were given that sec(T) = -10/7
if T were in quadrant 1, you would get sec(T) = 10/7.
since sec(T) = 1/cos(T), then solve for:
1/cos(T) = 10/7
solving for cos(t) results in:
cos(T) = 7/10.
use your calculator to find T to get:
T = 45.572996
that's in quadrant 1.
in quadrant 2, the angle would be (180 - 45.572996) which makes:
T = 134.427004

now that you have the angle in quadrant 2, you can check to see if you got the other values correctly.

sec(134.427004) = -1.428571429
sin(134.427004) = .7141428429
cot(134.427004) = -.9801960588

to confirm that these values are the same as the values you previously calculated, do the following:

previously you got or were given:
sec(T) = -10 / 7
sin(T) = sqrt(51)/10
cot(T) = -7/sqrt(51)

put -10/7 in your calculator and you get -1.428571429 which matches sec(134.427004) so that answer is good.

put sqrt(51)/10 in your calculator and you get .7141428429 which matches sin(134.427004) so that answer is good.

put -7/sqrt(51) in your calculator and you get -.9801960588 which matches cot(134.427004) so that answer is good.

everything checks out and your answer is:

sin(T) = sqrt(51) / 10

cot(t) = -7 / sqrt(51)

a picture of your angle is attached.