Courtney T.

Second-year College Student

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Pre-Calculus

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Question:

Simplify the following: $$sin(\theta) + cot(\theta)cos(\theta)$$

Courtney T.

Answer:

First, we want to change as much as we can into $$sin$$ and $$cos$$. Cotangent is the ratio $$cos(\theta)/sin(\theta)$$, so we can replace $$cot(\theta)$$ with $$cos(\theta)/sin(\theta)$$. That gives us $$sin(\theta) + [cos(\theta)/sin(\theta)][cos(\theta)/1]$$. In our second term, we can see that there are two cosines in the numerator with can be multiplied together to give us $$sin(\theta) + cos^2(\theta)/sin(\theta)$$. Next we have to make sure both terms have the same denominator. In this case, it is simple: multiply $$sin(\theta)$$ by $$sin(\theta)/sin(\theta)$$. This doesn't change the value of the term because our multiplier is technically equal to one. Our expression should now look like this: $$sin^2(\theta)/sin(\theta)+cos^2(\theta)/sin(\theta)$$ We can easily make this one term by combining the numerators over the common denominator like so: $${[sin^2(\theta)+cos^2(\theta)]}/sin(\theta)$$. Using the trigonometric identity $$sin^2(\theta)+cos^2(\theta)=1$$, we can replace the numerator with 1. We now have $$1/sin(\theta)$$. Simplifying one last time, our final answer is $$csc(\theta)$$.

US Government and Politics

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Question:

What are the differences between expressed, implied, and reserved powers, and give an example/situation of each.

Courtney T.

Answer:

There are several differences between the powers. Express powers are expressly stated in the Constitution: The President has the power to veto bills. Implied powers stem largely from the "necessary and proper clause" (Article 1, Section 8) of the Constitution. These powers are not stated outright, but are implied through interpretation of the Constitution, such as the creation of an interstate highway system to regulate commerce, one of the federal government's expressed powers. Finally, reserved powers are powers left to the state and can involve things such as developing a school system.

Algebra

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Question:

Using the following two functions, solve the given composites. $$f(x$$)=$$x/4$$ $$g(x)$$=$$-5x+3$$ Solve: $$f(g(x))=-3$$ and $$g(f(x))=-7$$

Courtney T.

Answer:

A composite function such as $$f(g(x))$$ means that the function $$g(x)$$ is placed inside the function $$f(x)$$. For each $$x$$ in $$f(x)$$, the function for $$g(x)$$ replaces it. So to solve $$f(g(x))$$, we would first have to find all values of $$x$$ and replace them with $$g(x)$$, or $$-5x+3$$. That would look like this: $$(-5x+3)/4$$. Given $$f(g(x))=-3$$, we solve for x, which should give us $$x=3$$. Doing $$g(f(x))$$ is very similar, but the equations are flipped. This time, we put $$f(x)$$ in $$g(x)$$. So replace every $$x$$ in $$-5x+3$$ with $$x/4$$ and we get $$(-5x/4)+3$$. Given $$g(f(x))=-7$$, we again solve for $$x$$, this time giving us $$x=8$$.

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