# Thread: Sin and cos question?

1. ## Sin and cos question?

I really don't get this problem. The angle seems as though it is 45 degrees. So for a) do I put it as this sin(θ + 360) then sin(45 + 360) = 0.7071? Or what am I suppose to do? Please help me with that problem and also if you could do b) also as an example that would be great. Thank you so much.

2. Originally Posted by florx

I really don't get this problem. The angle seems as though it is 45 degrees. So for a) do I put it as this sin(θ + 360) then sin(45 + 360) = 0.7071? Or what am I suppose to do? Please help me with that problem and also if you could do b) also as an example that would be great. Thank you so much.
Hi florex,

it will make more sense if you put (0,0) at the intersection of the axes.
Then notice

$sin\theta=a=y\ co-ordinate$

Hence $sin(360+\theta)=a$ as that is around the circle once.

For (b), move 180 degrees anticlockwise from the starting point,
brings the point a length 'a" under the x-axis to the left.

Also, given that $cos\theta$ gives the x co-ordinate, the others can be evaluated similarly.

3. You are not supposed to find/guess the actual angle.

If $\sin \theta = a$ then $\sin (360+ \theta) = a$ as in this, $\theta = 360+\theta$ as both angles will be in the first quadrant.

If $\sin \theta = a$ then $\sin (180+ \theta) = -a$ as $= 180+\theta$ is negative as it is in the 3rd quadrant.

Do you follow?

4. Thank you so much for explaining that to me. I get it now thanks for your teachings.

5. As a confirmation, is the answer c) is to move 90 degrees clockwise so the a should be in the 4th quad and thus the a would be -a?

6. Originally Posted by florx
As a confirmation, is the answer c) is to move 90 degrees clockwise so the a should be in the 4th quad and thus the a would be -a?
No florx,

You must start at $90^o$ and move $\theta$ degrees clockwise from there.

Imagine placing the triangle against the y-axis instead of the x-axis,
with the hypotenuse in the first quadrant.

Then notice

$cos(90-\theta)=a,\ the\ x\ co-ordinate$

7. Oh I see talking about now. As for d) we would start at 180 degrees and then move θ degrees clockwise from there which would make the triangle in the 2nd quad. However I am still stuck on the sin and cos sign part. I know cos means the x coordinates and the sin means the y coordinates. But look at what the paragraph says at the end, "Evaluate the follow expressions in terms of a." And also what you mean by "the x co-ordinate"?

8. Originally Posted by florx
Oh I see talking about now. As for d) we would start at 180 degrees and then move θ degrees clockwise from there which would make the triangle in the 2nd quad. However I am still stuck on the sin and cos sign part. I know cos means the x coordinates and the sin means the y coordinates. But look at what the paragraph says at the end, "Evaluate the follow expressions in terms of a." And also what you mean by "the x co-ordinate"?
Hi florx,

Have a look at the original diagram.
If you give the horizontal side of the triangle the length "b", then

$cos\theta=b=x\ co-ordinate$

Normally, we have $cos(\theta)=\frac{adjacent}{hypotenuse}$

but the hypotenuse is 1 here, so $cos(\theta)=adjacent=x$

Therefore $cos\theta=b$

$sin\theta=a$

Now, when you move the triangle as suggested, to find $cos(90-\theta)$
you bring a vertical line down from the point on the circle circumference onto the x-axis.
Notice that the distance from this point on the x-axis to (0,0) is 'a", hence

$cos(90-\theta)=a$

The sign (+ or -) of sin and cos are the signs of the y and x co-ordinates.

If x is +, then $cos(angle)$ is +
If x is -, then $cos(angle)$ is -
If y is +, then $sin(angle)$ is +
If y is -, then $sin(angle)$ is -

The x co-ordinate is the horizontal position of the point on the circumference, left or right of (0,0).
The y co-ordinate is the vertical position of the point on the circumference.