State the value of Θ, using both a counterclockwise and a clockwise rotation.

Given each point P(x, y), lying on the terminal arm of angle Θ:

i) state the value of Θ, using both a counterclockwise and a clockwise rotation.

ii) determine the primary trigonometric ratios

10c) P (-1, 0)

So, this is what I did:

x = -1, y = 0, r = 1

sinΘ = 0/1

sinΘ=0

cosΘ = -1/1

cosΘ = -1

tanΘ = 0/1

tanΘ = 0

And then to determine the counterclockwise and clockwise ratios (this is the part I got wrong), I did...

cosΘ = -1

Θ = 180

180+180 = 360

180-180 = 0

However, this is wrong according to the textbook. It says the answer is: 180 and -180 degrees. Can someone explain how they got that? Thanks!

Re: State the value of Θ, using both a counterclockwise and a clockwise rotation.

Quote:

Originally Posted by

**eleventhhour** Given each point P(x, y), lying on the terminal arm of angle Θ:

i) state the value of Θ, using both a counterclockwise and a clockwise rotation.

ii) determine the primary trigonometric ratios

10c) P (-1, 0)

Because I am a Pythagorean, it is against my religion to use degrees. I must use *numbers* as measures.

Therefore $\pi~\&~-\pi$ both represents the point $(-1,0)$.

$\pi$ is the counter-clockwise measure and $-\pi$ is the clockwise measure.

For the point $\left(\dfrac{-\sqrt 3}{2},\dfrac{1}{2}\right)$ is $\dfrac{2\pi}{3}$ is the counter-clockwise measure and $-\dfrac{5\pi}{3}$

is the clockwise measure .

Re: State the value of Θ, using both a counterclockwise and a clockwise rotation.

That's not helpful in the slightest. This is just grade 11 math - we use degrees.

Can someone else help (on a grade 11 level)?

Re: State the value of Θ, using both a counterclockwise and a clockwise rotation.

Quote:

Originally Posted by

**eleventhhour**

And then to determine the counterclockwise and clockwise ratios.

The question you gave at the start was,

i) state the value of Θ, using both a counterclockwise and a clockwise rotation.

Are these two questions the same?

I usually think of cosine/sine by imagining the unit circle. A positive input follows the unit circle upwards and to the left, while a negative input follows the unit circle downwards and to the left. All this question wants you to do is find the positive and negative $\theta$ which describe the same point. Which explains where their answer, $cos(\theta)=-1$ for $\theta=180$ and $\theta=-180$, comes from.

I'm not sure how your text is telling you to get the counterclockwise and clockwise rotations though so I can't help out more.

Re: State the value of Θ, using both a counterclockwise and a clockwise rotation.

Quote:

Originally Posted by

**eleventhhour** That's not helpful in the slightest. This is just grade 11 math - we use degrees.

Can someone else help (on a grade 11 level)?

No responsible educational authority allows mathematics standards to use degrees.

Shame on wheresoever you are in Canada. With that old fashion idea you are disadvantaged at higher mathematics.

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Re: State the value of Θ, using both a counterclockwise and a clockwise rotation.

Quote:

Originally Posted by

**bkbowser** I'm not sure how your text is telling you to get the counterclockwise and clockwise rotations though so I can't help out more.

Oh my, surely you know about a unit circle???

Attachment 30886

Don't you know clock-wise from counter-clockwise?

Re: State the value of Θ, using both a counterclockwise and a clockwise rotation.

Conventionally, one arm of an angle on the unit circle is conceived to be FIXED along the line running from the origin to the point (0, 1).

Also conventionally, an angle is conceived to be positive if measured counter-clockwise and negative if measured clockwise.

So to move the non-fixed arm from (0, 1) to (0, -1), that arm must be moved half way around the unit circle whether moved positively or negatively. In degrees, moving through a half circle counter-clockwise is PLUS 180 degrees, and moving through a half-circle clockwise is MINUS 180 degrees. The fact that moving through a half circle or a circle involves an angle of the same minimum magnitude whether in the positive or negative directions is an important one to know. Does this answer your immediate question?

Now not being a mathematician myself, I do not feel contempt for non-mathematicians, I believe that they deserve to get those rudiments of mathematics that are necessary for practical life in a technological society and as much more mathematics as they find desirable for their own intellectual development. In many practical applications, understanding and using degrees is necessary. But for everyone who wants or needs more than the rudiments of mathematics, they need to understand that degrees are not the most convenient or even the most intuitive way of measuring angles: radians are. You should be able to go from angle to radian measure without hesitation. An answer in radians should be just as comprehensible to you as an answer in degrees if you intend to be able to comprehend the mathematics that, in my opinion, are required to be a well educated person in the modern world. (Remember that I do not say this as a mathematician or a teacher.) Now it may be that you have not been introduced to radian measure yet. It is simplicity itself. 180 degrees equal pi radians. That's it in terms of mechanics. So 90 degrees is pi/2 radians, and 360 degrees is 2pi radians, etc.

Re: State the value of Θ, using both a counterclockwise and a clockwise rotation.

Quote:

Originally Posted by

**Plato** No responsible educational authority allows mathematics standards to use degrees.

Shame on wheresoever you are in Canada. With that old fashion idea you are disadvantaged at higher mathematics.

They could just be starting trigonometry and will get to radian measure soon. In Australia basic trigonometry is learnt before learning about radians.