Using Function Properties to Evaluate Related Points

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These problems are about using function properties to evaluate trigonometric functions.

1 : Suppose $\cos (\alpha )=\frac {\sage {p1}}{\sage {q1}}$ .

(a): What is $\cos (-\alpha )$ ? $\answer {\sage {ans1a}}$
(b): What is $\sec (\alpha )$ ? $\answer {\sage {ans1b}}$

2 : Suppose $\sin (\beta )=\frac {\sage {p2}}{\sage {q2}}$ .

(a): What is $\sin (-\beta )$ ? $\answer {\sage {ans2a}}$
(b): What is $\csc (2\pi +\beta )$ ? $\answer {\sage {ans2b}}$

3 : Suppose $\tan (\alpha )=\frac {\sage {p3}}{\sage {q3}}$ .

(a): What is $\tan (-\alpha )$ ? $\answer {\sage {ans3a}}$
(b): What is $\cot (\alpha )$ ? $\answer {\sage {ans3b}}$
(c): What is $\tan (\alpha - 4\pi )$ ? $\answer {\sage {ans3c}}$

4 : Suppose $\sin (t)=\frac {\sage {p41}}{\sage {q41}}$ and $\cos (t)=\frac {\sage {p42}}{\sage {q42}}$

(a): What is $\sage {coef41}\sin (t)+\sage {coef42}\cos (-t)$ ? $\answer {\sage {ans4a}}$
(b): What is $\sage {coef43}\sin (-t+6\pi )+\sage {coef44}\cos (-t)$ ? $\answer {\sage {ans4b}}$
(c): What is $\tan (t)-\cot (-t)$ ? $\answer {\sage {ans4c}}$