1 : $\alpha , \beta , \text { and } \gamma$ are angles in quadrants 1, 2, and 3, respectively, such that: $$\begin{align*} \tan \alpha &= \sage{DispAlpha} \\ \tan \beta &= \sage{DispBeta} \\ \cos \gamma &= \sage{DispGamma} \end{align*}$$

Assume also that $0 < \alpha , \beta , \gamma < 2\pi$. Evaluate the following.

(a)

$\sin (2\gamma ) = \answer {\sage {Ans1a}}$

(b)

$\cos (2\beta ) = \answer {\sage {Ans1b}}$

(c)

$\tan (2\alpha ) =\answer {\sage {Ans1c}}$

(d)

$\csc (2\beta ) = \answer {\sage {Ans1d}}$

(e)

$\sec (2\gamma ) = \answer {\sage {Ans1e}}$

(f)

$\sin (4\gamma ) = \answer {\sage {Ans1f}}$

(g)

$\cos (2\gamma +2\beta ) = \answer {\sage {Ans1g}}$

2 : Write the following as algebraic expressions.

(a)

$\sin (2\arcsin (\sage {O2})) = \answer {\sage {Ans2a}}$

(b)

$\tan (2\arcsin (\sage {O2})) = \answer {\sage {Ans2b}}$

3 : $\alpha , \beta , \text { and } \gamma$ are angles in quadrants 1, 2, and 3, respectively, such that: $$\begin{align*} \tan \alpha &= \sage{Disp2Alpha} \\ \tan \beta &= \sage{Disp2Beta} \\ \cos \gamma &= \sage{Disp2Gamma} \end{align*}$$

Assume also that $0 < \alpha , \beta , \gamma < 2\pi$. Evaluate the following.

(a)

$\cos \left (\frac {\gamma }{2}\right ) = \answer {\sage {Ans3a}}$

(b)

$\tan \left (\frac {\gamma }{2}\right ) = \answer {\sage {Ans3b}}$

(c)

$\sin \left (\frac {\beta }{2}\right ) = \answer {\sage {Ans3c}}$

(d)

$\sec \left (\frac {\alpha }{2}\right ) = \answer {\sage {Ans3d}}$