Suppose θ is an angle in standard position whose terminal side lies in Q II. If sin θ =
13
12
, find sec θ.
To find sec θ, we need to use the reciprocal identity of sine. The reciprocal identity for sine states that sec θ is equal to 1 divided by cosine θ.
First, we need to find the value of cosine θ. Since θ is in Q II (quadrant II), we know that sine is positive (13/12 is positive) and cosine is negative in this quadrant.
To find the value of cosine θ, we can use the Pythagorean identity, which states that sin^2 θ + cos^2 θ = 1.
Given that sin θ = 13/12, we can substitute this value into the Pythagorean identity:
(13/12)^2 + cos^2 θ = 1
169/144 + cos^2 θ = 1
cos^2 θ = 1 - 169/144
cos^2 θ = 144/144 - 169/144
cos^2 θ = -25/144
Since θ is in Q II, the cosine value is negative. Therefore, cos θ = -√(25/144) = -5/12.
Now we can find sec θ using the reciprocal identity:
sec θ = 1/cos θ = 1/(-5/12) = -12/5.
Therefore, sec θ = -12/5.