How do you use fundamental identities to find the values of the trigonometric values given angle in quadrant III, such that sec(x) = -3?

Answer 1

Please see the explanation.

Use the identity #sec(x) = 1/cos(x)#:
#cos(x) = -1/3#
Use the identity #sin(x) = +-sqrt(1 - cos^2(x))#
Because we are given that x is in the third quadrant, we know that the sine function is negative so we remove the #+#sign:
#sin(x) = -sqrt(1 - cos^2(x))#
Substitute #(-1/3)^2" for "cos^2(x)#
#sin(x) = -sqrt(1 - (-1/3)^2)#
#sin(x) = -sqrt(1 - 1/9)#
#sin(x) = -sqrt(8/9)#
#sin(x) = (-2sqrt(2))/3#
Use the identity #csc(x) = 1/sin(x)#
#csc(x) = (-3sqrt(2))/4#
Use the identity #tan(x) = sin(x)/cos(x)#
#tan(x) = ((-2sqrt(2))/3)/(-1/3) = 2sqrt(2)#
Use the identity #cot(x) = 1/tan(x)#:
#cot(x) = 1/(2sqrt(2)) = sqrt(2)/4#
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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