$\sec \theta = k,\,|k| \ge 1$ and that $\theta $ is obtuse, express in terms of $k$:

(1) $\cos \theta $

(2) ${\tan ^2}\theta $

(3) $\cot \theta $

(4) $\csc \theta $

My attempt:

(1) $\cos \theta = \dfrac{1}{k}$

(2) ${\tan ^2}\theta = {k^2} - 1$

(3) $\dfrac{-1}{\sqrt{k^2 - 1}}$

(4) $\csc \theta = \dfrac{k}{\sqrt{k^2 - 1}}$

My answers to (1) and (4) are wrong, can anyone tell me why this is? Isn't cosine meant to be negative if the angle lies in the second quadrant (i.e. obtuse)? And shouldn't sine (and therefore csc) be positive in the obtuse angle range?



Your answer to (1) is correct: the cosine is the reciprocal of the secant. So are your answers to (2) and (3).

The cosecant should be positive in the second quadrant; therefore the answer is $$ \csc\theta= \frac{-k}{\sqrt{k^2-1}}, $$ which is positive, since $k$ is negative.


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