For the following exercise,

Find the exact value of each trigonometric function.

$$\sin \frac{\pi}{2}$$

Solve each equation for x, where x in [0, 2pi].

a) sec x csc x - 2 csc x = 0

b) 3 sec ^2 x - 4 = 0

c) 2 sin x sec x - 2 square root of 3 sin x = 0

d) 2 cot x + sec^2 x = 0

e) cot x csc^2 x = 2 cot x

f) 3 tan^3 x - tan x = 0

In this exercise, use a calculator to find the value of the trigonometric function to four decimal places.

$$\tan 3.4$$

For the following exercise,

Find the exact value.

$$\sin \left(\frac{7 \pi}{8}\right)$$

What are the products of reaction of (E)-3-methyl-3-hexene with each of the reagents in the indicated problem?

For the following exercise,

Find the exact value of each trigonometric function.

$$\cos \frac{\pi}{2}$$

Verify that each equation is an identity.

$$\cos 2 \theta=\frac{2-\sec ^2 \theta}{\sec ^2 \theta}$$

Karl ralses his hand and Informs Ms. Rhombi that he has discovered another property related to the angles of an inscribed quadrilateral. He claims that the measure of any exterior angle of the quadrilateral is equal to the measure of the opposite interior angle in the quadrilateral. In other words, $m \angle A W Z=m \angle Y$. Explain Karl's reasoning.

A car under surveillance accelerates at a constant rate from 0 mph to 60 mph in 30 sec. How far has it traveled after 30 sec?

Show how you would synthesize 2-phenylhex-3-yn-2-ol, starting with acetophenone $\left(\mathrm{PhCOCH}_{3}\right)$ and any other reagents you need. (“2-ol” means there is an OH group on C2.)

Integrate

$$\int \sec x \tan x d x$$

Rewrite $\dfrac{1}{n}\sec^2\left(1 + \dfrac{1}{n}\right) + \dfrac{1}{n}\sec^2\left(1 + \dfrac{2}{n}\right) + \dfrac{1}{n}\sec^2\left(1 + \dfrac{3}{n}\right) + \ldots + \dfrac{1}{n}\sec^2\left(1 + \dfrac{n}{n}\right)$ using sigma notation. Do not evaluate.

Find $y^{\prime}$.

$$y=e^{x \sec x}$$

For the following exercise,

Let $f(x)=\frac{3}{5} \cos (6 x)$.

What is the smallest possible value for $f(x)$?