Half Angle Identities

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This collection of problems explores the power reducing formulas.

1 : Suppose $\theta$ is an angle in quadrant 4. Which quadrants could $\frac {\theta }{2}$ be in?

Quadrant 1 Quadrant 2 Quadrant 3 Quadrant 4

Which quadrants could $\frac {\theta }{4}$ be in?

Quadrant 1 Quadrant 2 Quadrant 3 Quadrant 4

1.1 :

Suppose $\phi$ is an angle in quadrant 1. Which quadrants could $\frac {\phi }{2}$ be in?

Quadrant 1 Quadrant 2 Quadrant 3 Quadrant 4

Which quadrants could $\frac {\phi }{4}$ be in?

Quadrant 1 Quadrant 2 Quadrant 3 Quadrant 4

2 : $\alpha , \beta , \text { and } \gamma$ 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): $\cos \left (\frac {\gamma }{2}\right ) = \answer {\sage {Ans2a}}$
(b): $\tan \left (\frac {\gamma }{2}\right ) = \answer {\sage {Ans2b}}$
(c): $\sin \left (\frac {\beta }{2}\right ) = \answer {\sage {Ans2c}}$
(d): $\sec \left (\frac {\alpha }{2}\right ) = \answer {\sage {Ans2d}}$

3 : $\alpha , \beta , \text { and } \gamma$ are angles in quadrants 1, 2, and 3, respectively, such that: $\begin{align*} \tan \alpha &= \sage{Disp3Alpha} \\ \tan \beta &= \sage{Disp3Beta} \\ \cos \gamma &= \sage{Disp3Gamma} \end{align*}$

This time we also that $2\pi < \alpha , \beta , \gamma < 4\pi$ . 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): $\sec \left (\frac {\alpha }{2}\right ) = \answer {\sage {Ans3d}}$

4 : $\theta$ is an angle in quadrant 3 such that $-\pi < \theta < -\frac {\pi }{2}$ and such that $\cos (\theta )=\sage {Disp4}$ . Calculate the following.

(a): $\sin \left (\frac {\theta }{2}\right ) = \answer {\sage {Ans4a}}$
(b): $\cos \left (\frac {\theta }{2}\right ) = \answer {\sage {Ans4b}}$
(c): $\tan \left (\frac {\theta }{2}\right ) = \answer {\sage {Ans4c}}$