Verify each identity.

$$\tan \frac { \alpha } { 2 } = \frac { \tan \alpha } { \sec \alpha + 1 }$$

$$\text { Find an isomorphism from } Z _ { 12 } \text { to } Z _ { 4 } \oplus Z _ { 3 }$$

Evaluate the expression.

$$\frac { x } { 5 }$$

when x = 30

Find the vertex of each parabola. focus: (-2, -1); directrix: y = -6

Use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. $y=\frac{1}{\cot x+1}+\frac{1}{\tan x+1}$

use an inverse matrix to solve (if possible) the system of linear equations. {4x - 2y = -10 , -19x + 9y = 47

Use Cramer's rule to solve each system.

$$\left\{ \begin{array} { r r } { 2 x + 2 y + 3 z = 10 } \\ { 4 x - y + z = - 5 } \\ { 5 x - 2 y + 6 z = 1 } \end{array} \right.$$

Find

$$\sin \frac { x } { 2 } ,\cos \frac { x } { 2 }$$

, and

$$\tan \frac { x } { 2 }$$

from the given information.

$$\cos x = - \frac { 4 } { 5 } , \quad 180 ^ { \circ } < x < 270 ^ { \circ }$$

Explain what transformations you would need to apply to the graph of

$$y=f\left(x\right)$$

to graph each function. a)

$$y=3f\left(x\right)-1$$

b)

$$y=f\left(x-2\right)+3$$

c)

$$y=f\left(2x\right)-5$$

d)

$$y=-f\left(\frac{1}{2}x\right)-2$$

e)

$$y=\frac{2}{3}f\left(x+3\right)+1$$

f)

$$y=4f\left(-x\right)-4$$

Simplify: 3 5/6 - 1/3

Find the indicated term of the arithmetic sequence.

$$a_1 = 4, d = 5; a_{15}$$

$$14, 18, 22, 26, . . .;a_{20}$$

$$45, 41, 37, 33, . . .;a_{18}$$

Give an example of a function with local maxima and minima at an infinite number of points.

State the criteria that must be met for an experiment to be a binomial experiment.

Explain how expected values and probabilities are related.