Question

Quick question:
How to evaluate AX = (cos 6)(sin 12) / (sin 162)

Answer 1

aum!!

Answer 2

helpp please

Answer 3

also, why is it that sin 162 =sin 18?

Answer 4

*teacher said that might help

Answer 5

does that mean that sin x = sin 180-x

Answer 6

The reference angle is 18 degrees.
Sine is positive in both the first and second quadrant.
So sin(162) = sin(18)

Answer 7

Yes. sin(x) = sin(180-x)

Answer 8

ok thanks. so would it help to rewrite sin 162 as sin 18 in the problem?

Answer 9

i need to solve it in this form: (cos x)(sin y)(csc z)

Answer 10

(cos 6)(sin 12) / (sin 162) = (cos 6)(sin 12) / (sin 18) = (cos 6)(sin 12)(csc 18)

Answer 11

THANKS!!!!!! would this be wrong?: (cos 6)(sin 12)(csc 162)?

Answer 12

No. It won't be wrong. But it is better to simplify as much as possible.

Answer 13

so the lower number is considered more simplified?

Answer 14

Yeah. It is better to write sin(30) instead of sin(390)

Answer 15

Ok thanks. But i can technically write sin 30 as sin 750?, or for that matter, sin 360x +30 (where x is an integer)?

Answer 16

Yes. calculators will give the same value for sine of all those angles. But if you can express it as an angle, preferably in the first quadrant if possible, then that would be preferred.

Answer 17

waiiiit. What do you mean by quadrants? How is this connected to the coordinate grid? can there be negative sines? sines between 180 and 360?

Answer 18

If sin( some big angle ) can be equivalently written as sin( 0 <= angle <= 90 ) then the latter is preferred. First quadrant angle means angle from (0,90).

Answer 19

what about angles above 90 but below 360? it doesn;t make sense to me in the sense of a right triangle, as i have learned sines

Answer 20

And yes, angles for negative sines will be in the third or fourth quadrant.

Answer 21

sin -30 = sin 30???

Answer 22

No. Sin -30 is in the fourth quadrant where sine is negative. So sin(-30) = -sin(30).

Answer 23

sin(-30) is same as sin(-30+360) or sin(330).

Answer 24

ok. I am thinking about this whole sine thing in terms of a right triangle. Is that the wrong way to be perceiving it? because i am not getting how a sin (opp/hyp) can be negative

Answer 25

The three angles of a triangle add to 180 degrees. Therefore no one angle can be 180 degrees or more. So for angles greater then 180 don't use the triangle definition. But of you know about reference angles then you can reduce sine of ANY angle to sine of an angle from 0 to 90 degrees.

Answer 26

*if you know*

Answer 27

Reference angles? Sorry, I've never heard of it. Is it a hard concept?

Answer 28

Not really. But rather than explain the concept by slowly typing a line at a time here you can type reference angle in YouTube or in Google search and read about it.

Answer 29

alright got it

Answer 30

Example. Find sin(1230).
You can subtract as many multiples of 360 degrees as possible and the sine value will be unaffected. 1230-360 = 870
870-260 = 510
510-360=150

150 degrees is in the second quadrant where sine is positive.

A line that makes 150 degrees with the positive x-axis makes 30 degrees with the negative x-axis. The reference angle is 30 degrees. Since sine is positive in the second quadrant, sin(150) = sin(30).

Therefore, sin(1230) = sin(870) = sin(510) = sin(150) = sin(30) = 1/2

Answer 31

ok, so once you have the reference angle, you simply determine whether it is negative or positive based on quadrant?

Answer 32

correct. reference angle is the angle the line makes with the positive x-axis if the line is in the first or fourth quadrant and the angle it makes with the negative x-axis if the line is in the second or third quadrant.

Answer 33

GOT it!!!!!!!!!!!! THANKS!!!!!!!! but wait. Are cos,tan,csc,sec,and cot different?

Answer 34

Find cos(-30).
The reference angle here is 30 degrees.
Since cosine is positive in the fourth quadrant, cos(-30) = cos(30) = \sqrt(3) / 2

Answer 35

are refernce angles only for sines?

Answer 36

For all trigonometric functions.

Answer 37

becuase sin -30 is sin 330 correct?

Answer 38

what i'm asking is why is it that cos 30 = cos -30 while sin doesnt behave this way

Answer 39

sin(-30) = sin(330)

But if I was asked to evaluate sin(-30) I would not go that route. I will say the reference angle for -30 degrees is +30 degrees. Since -30 lies in the fourth quadrant, sine is negative there. Therefore, sin(-30) = -sin(30) = -1/2.

Answer 40

Remember the acronym ASTC (or All Students Take Calculus)
A: All trig functions are positive in the first quadrant
S: Sine (and therefore, csc) are positive in the second quadrant
T: Tan (and hence, cot) are positive in the third quadrant
C: Cos (and hence, sec) are positive in the fourth quadrant.

Answer 41

ohhhhhhhhhhhhhhhh. I get it now. that makes a lot of sense thanks! cosine is positive in the fourth quadrant so cos -30 = cos 30!

Answer 42

right?

Answer 43

Yes. In general cos(x) = cos(-x)

cosine and secant are called EVEN functions. Changing x to -x does not affect them. These functions are symmetrical with respect to the y-axis when drawn because they have the same y-value at x and -x.