This is my first calculus class and I am lost on how to answer these. The book isn't giving me a clear explanation for help. Any help someone could provide would be great!

1. Use a reference triangle to find the given angle.
cos -1 (1/2) give an exact answer in terms of pi

2. Express In 0.25 in terms of In 2 and In 3. Simplify your answer. Use integers or fractions for any number in the answer.

3. Determine whether the function is even, odd or neither.
f(x)=4x^5 + 3x^3

Thank you!!!!

1. Use a reference triangle to find the given angle.

cos -1 (1/2) give an exact answer in terms of pi

adjacent/hypotenuse = 1/2
well, if you know about 30,60,90 triangle the answer is immediate.
sin 30 degrees = 1/2
cos 60 degrees = 1/2
60 degrees = pi/3 radians

2. Express In 0.25 in terms of In 2 and In 3. Simplify your answer. Use integers or fractions for any number in the answer.

ln (1/4) = ln 1 - ln 4 = 0 - ln 4

-ln 4 = -2 ln 2

3. Determine whether the function is even, odd or neither.

f(x)=4x^5 + 3x^3

For negative x, this function is negative of the corresponding value for positive x
Therefore it is ODD
In general odd powers of x are odd and even powers of x are even
eg
f(x ) = +x
f(-x) = -x odd

f(x ) = + x^2
f(-x) = + x^2 even

Sure, I'd be happy to help you understand how to solve these problems!

1. To find the angle using a reference triangle, we need to use the inverse cosine function. In this case, we have cos^(-1)(1/2). The cosine function represents the ratio of the adjacent side to the hypotenuse in a right triangle. Since the cosine of an angle is 1/2, we need to find the angle whose adjacent side is 1 and hypotenuse is 2.

To do this, we can draw a right triangle where the adjacent side is 1 and hypotenuse is 2. Since the angle we're looking for is acute, it will be the same as the reference angle in the first quadrant of the unit circle. Therefore, the opposite side will be √(2^2 - 1^2) = √3, applying the Pythagorean theorem.

Now, we can find the angle using the reference triangle. The inverse cosine function gives us the angle in radians. So, cos^(-1)(1/2) = √3/2 in terms of π. This means the angle is √3/2 radians, or approximately 60 degrees.

2. To express In(0.25) in terms of In(2) and In(3), we need to use logarithmic properties. First, we can rewrite 0.25 as 1/4, because 0.25 is equivalent to 1 divided by 4. Now we have to express 1/4 as a power of 2 and 3.

Since 2^(-2) = 1/4, we can rewrite 1/4 as 2^(-2). Similarly, 3^(-2) = 1/9, so we can also rewrite 1/4 as 3^(-2).

Now, we can express In(0.25) in terms of In(2) and In(3) using logarithmic properties. In(0.25) = In(2^(-2)) = -2 * In(2), and In(0.25) = In(3^(-2)) = -2 * In(3).

So, In(0.25) = -2 * In(2) = -2 * In(3).

3. To determine whether the function f(x) = 4x^5 + 3x^3 is even, odd, or neither, we need to check if it satisfies certain properties.

First, let's substitute -x for x in the function:
f(-x) = 4(-x)^5 + 3(-x)^3

Now, we simplify this expression:
f(-x) = -4x^5 - 3x^3

If f(-x) is equal to f(x), then the function is even. However, if f(-x) is equal to the negative of f(x), then the function is odd. If neither condition is met, then the function is neither even nor odd.

In this case, we can see that f(-x) is equal to -f(x), which means the function is odd.

I hope this helps you understand how to solve these problems! If you have any more questions or need further clarification, feel free to ask.