In order to find the exact value of we can use the half angle formula for sin, which is

.

This way we can plug in a value for alpha for which we know the exact value. is equal to divided by two, and so we can plug in for the alpha above.

The cosine of is .

Therefore our final answer becomes,

.

For any function , the first derivative .

Therefore, taking each term of :

To derive, use the power rule for derivatives.

Find the first derivative by taking the derivative of each term.

Take the derivative of .

Which of the following is equivalent to the following expression?

Recall our Pythagorean trig identity:

It can be rearranged to look just like our numerator:

So go ahead and change our original expression to:

Then recall the definition of cosecant:

So our original expression can be rewritten as:

So our answer is:

In order to find the derivative we will need to use the power rule on each term. The power rule states,

.

Applying this rule we get the following.

To find the derivative of this function, we simply need to use the Power Rule. The Power rule states that for each term, we simply multiply the coefficient by the power to find the new coefficient. We then decrease the power by one to obtain the degree of the new term.

For example, with our first term, , we would multiply the coefficient by the power to obtain the new coefficient of . We then decrease the power by one from 4 to 3 for the new degree. Therefore, our new term is . We then simply repeat the process with the remaining terms.

Note that with the second to last term, our degree is 1. Therefore, multiplying the coefficient by the power gives us the same coefficient of 8. When the degree decreases by one, we have a degree of 0, which simply becomes 1, making the entire term simply 8.

With our final term, we technically have

Therefore, multiplying our coefficient by our power of 0 makes the whole term 0 and thus negligible.

Our final derivative then is

To find the derviative of this equation recall the power rule that states: Multiply the exponent in front of the constant and then subtract one from the exponent.

We can work individually with each term:

Derivative of

is,

For the next term:

Derivative of :

So answer is:

Anything to a power of 0 is 1.

For the next term:

Derivative of :

Any derivative of a constant is .

So the first derivative of

is

By the Power Rule of derivatives, for any equation , the derivative .

With our function , where , we can therefore conclude that:

To figure out the horizontal component, set up an equation involving cosine, since that side of the implied triangle is adjacent to the 48-degree angle:

To solve for x, first find the cosine of 48, then multiply by 11:

To figure out the vertical component, set up an equation involving sine, since that side of the implied triangle is opposite the 48-degree angle:

to solve for y, just like x, first find the sine of 48, then multiply by 11:

Putting this in component form results in the vector

In vector addition, you simply add each component of the vectors to each other.

x component: .

y component: .

z component: .

The new vector is

.

To find the magnitude we use the formula,

Thus its magnitude is 5.