How do you determine whether the sign of a trigonometric function (sine, cosine, tangent) is positive or negative when dealing with half angles? Explain your reasoning and cite examples. Why do you think the half-angle identities include positive and negative options but the other identities don't seem to have this option built in?

This question is about:

sin(A/2) and cos(A/2)

First, how we know when we need to use the positive or negative signs?

Ok, this part is kinda intuitive:

First, you need to know the negative/positve regions for the sine and cosine function.

Cos(x) is positive between 270 and 90, and negative between 90 and 270.

sin(x) is positive between  0 and 180, and negative between 180 and 360.

Then we need to see at the half-angle and see in which region it lies.

If the half-angle is larger than 360°, then you subtract 360° enough times such that the angle lies in the range between (0° and 360°)

and: Tan(A/2) = Sin(A/2)/Cos(A/2)

So using that you can infer the sign of the Tan(A/2)

Now, why these relationships use the two signs?

Well... this is because of the square root in the construction of the relationships.

This happens because:

(-√x)*(-√x) = (-1)*(-1)*(√x*√x) = (√x*√x)

For any value of x.

so both -√x and √x are possible solutions of these type of equations, but for the periodic nature of the sine and cosine functions, we can only select one of them.

So we should include the two possible signs, and we select the correct one based on the reasoning above.

341

step-by-step explanation:

all you have to do is 9x31 (which equals 279) and then 20x31 (which equals 620). then you have to subtract. 620-279=341.

the smallest possible measurement of the width is 14 feet

step-by-step explanation:

let's assume length of parallelogram is 'l'

width of parallelogram is 'w'

the perimeter of a parallelogram must be no less than 40 feet

so, we get

we are given

length of parallelogram is 6 feet

so,

l=6

we can plug it

now, we can solve for w

subtract both sides by 12

divide both sides by 2

and we get

so,

the smallest possible measurement of the width is 14 feet