1) Determine the values of the trigonometric functions of the angle (smallest positive angle) if P is a point on the terminal side and is the coordinates of P are P( -1, -3).
2)Find the diameter of a pulley which is driven at 360 r/min by a belt moving at 40 ft/s.
3) State the quadrant in which the angle terminates and the signs of the sine, cosine, and tangent of the angle of 212 degrees.
The teacher never went over this and i'm at a complete loss.
Please help :[
Look:Originally Posted by Lollerpoppe
You have a right triangle, with sides of length 3, 1, and $\displaystyle \sqrt{10}$. Surely you can evaluate the trigonometric functions here (if not, start paying attention!). Just remember that these lengths are "directed," so you need to take into account the negative signs. For example,
Although for our angle $\displaystyle \theta$, $\displaystyle \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac1{\sqrt{10}} = \frac{\sqrt{10}}{10}$, the angle $\displaystyle \alpha$ made with the positive $\displaystyle x$-axis is in quadrant III, so the cosine is negative and we have $\displaystyle \cos\alpha = -\frac{\sqrt{10}}{10}$.
You should be able to do the rest in similar fashion.
If the pulley is rotating at $\displaystyle 360\text{ rad/min} = 6\text{ rad/s}$, then a point on the edge of the pulley will be traveling at $\displaystyle 6r\text{ ft/s}$, where $\displaystyle r$ is the radius of the pulley in feet. So we have $\displaystyle 6r = 40$.
These are the quadrants, along with the signs of the trigonometric functions in each:
A mnemonic that is sometimes used for the signs is:
All - All three (sin, cos, and tan) are positive in QI
Students - Only sin is positive in QII
Take - Only tan is positive in QIII
Calculus - Only cos is positive in QIV
Another way of looking at it is to use the unit circle:
By drawing a perpendicular to the $\displaystyle x$-axis, you can see that every point on the unit circle is of the form $\displaystyle (\cos\theta,\;\sin\theta)$, where $\displaystyle \theta$ is the angle between the positive $\displaystyle x$-axis and the radial line containing the point. Thus you can easily see that the sine is positive wherever $\displaystyle y$ is positive (quadrants I and II), and the cosine is positive wherever $\displaystyle x$ is positive (quadrants I and IV).
Adding/Changing something to this:
A mnemonic that is sometimes used for the signs is:
All - All six (sin, cos, tan, sec, csc, and cot) are positive in QI
Students - Only sin and csc is positive in QII
Take - Only tan and cot is positive in QIII
Calculus - Only cos and sec is positive in QIV
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