**BOTH+8+AND+9**+(8)+Given+an+angle+determine+the+sign+of+the+trig+ratios+(9)+Determine+the+values+of+the+trig+ratios+at+quadrantal+angles

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=Objective 8: Given an angle (__degrees__ or radians), determine the sign of the trig ratios. =

The hardest part of __learning__ the sign of trig ratios is understanding what the unit circle means in relation to a coordinate plane. To make the connection, we look at a regular coordinate plane with plotted points.



__**We can see that in**:__
.... quadrant 1the point plotted reads (2,3). The values are both positive. (X,Y) .... quadrant 2 the point plotted reads (-4,1). The values are (-X,Y). .... quadrant 3 the point plotted reads (-3,-5). The values are (-X, -Y). ... quadrant 4 the point plotted reads (3, -1). The values are (X, -Y).



On a unit circle, the signs follow the same rules as for a plotted points in a coordinate plane. As a coordinate point is simply (X,Y), with a unit cirlce (X,Y) ---> (cos, sin). A unit circle is a circle with its center-point on (0,0) and a radius of 1 (as pictured). The signs (positive or negative) for (cos, sin) follow those of a (X,Y) pair plotted in a coordinate plane. //The coordinate plane below states the sign of coordinates in different quadrants.// When talk about (cos, sin), we can use a coordinate plane. However, for tangent, we look at rules of division. When dividing two positive numbers, the quotient is also positive. The quotient is also positive when dividing two negative numbers. Therefore, the tangent is positive in both Quad. 1 and 4. The other two quadrants involve one negative and one positive number in the coordinate pair. The result is therefore still negative [tangent negative in Quad. 2 and 3).
 * TANGENT:**


 * || Quadrant 1 || Quadrant 2 || Quadrant 3 || Quadrant 4 ||
 * cos || +  ||  -  ||  -  ||  +  ||
 * sin || +  ||  +  ||  -  ||  -  ||
 * __tan__ || +  ||  -  ||  +  ||  -  ||

Examples:**
 * For csc, sec, and cot, the sign in each quadrant is the same as its inverse. They are not the focus of this objective. For example, csc is the inverse of sin. If the sin is positive 1/2 (as it is in quadrant 1 and 2), then the csc is 2/1 (inverse) but still has the same sign of positive. This rule follows for all inverses. The inverse does not affect the sign.**
 * What is the sign of sin in quadrant 2?
 * //How to figure this out?//
 * Think of quadrant 2. All values have a signs of (-X, Y). So, (cos, sin) is changed to (-cos, sin)
 * Answer: Positive +
 * Check answer in chart [[image:precalculusnwr7/checkmark.gif]]


 * What quadrant is (1/2, -radical 3/2) in?
 * //How to figure this out?//
 * The Y is negative so looking at our coordinate plane Y's are negative in Quad. 3 and 4. Next, we look at the X value which is positive. This means the coordinate pair reads: (X, -Y) or (cos, -sin). The X value is positive in Quad. 1 and 4. Therefore, the point is in Quad. 4.
 * Answer: Quad. 4
 * Check answer with unit circle [[image:precalculusnwr7/checkmark.gif]]
 * What is the sign of cosine at 150 __degrees__on the unit circle?
 * //How to figure this out?//
 * What quadrant is 150 __degrees__ in? It is in quad. 2. Therefore, the coordinate pair reads (-X, Y) or (-cos, sin).
 * Answer: negative cosine
 * Check answer in chart [[image:precalculusnwr7/checkmark.gif]]

[[image:precalculusnwr7/bullet_11.gif width="28" height="28"]][[image:precalculusnwr7/bullet_11.gif width="28" height="28"]][[image:precalculusnwr7/bullet_11.gif width="27" height="27"]]
=Objective 9: Determine the values of trig ratios at quadrantal angles.=

__Quadrantal angle__: "an angles which has its terminal side coinciding with a coordinate axis"
 * All quadrantal angles listed in first column of chart

__How to determine the trig ratios?__
 * For sin: sin is the Y part of the coordinate so using a unit circle, look at vertical distance/rise (Y value)
 * For cos: cos is the X part of the coordinate so using a unit circle, look at horizontal distance/run (X value)
 * For tan: tangent is sin/cos, so take coordinate pair at quadrantal angle (X, Y) or (cos, sin) and divide Y/X or sin/cos
 * For csc: (inverse of sin or 1/sin), so take values of sin and put 1/value at quadrantal angle. (either equal zero or undefined [if fraction with denominator of zero])
 * For sec: (inverse of cos or 1/cos), so take values of cos and put 1/value at quadrantal angle. (either equal zero or undefined [if fraction with denominator of zero])
 * For cot: (inverse of tan or 1/tan or cos/sin or X/Y), so take tangent and switch denominator and numerator (or do X/Y OR cos/sin)

//This chart organizes the values of all trig functions at quadrantal angles://


 * [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci19.gif align="absBottom"]] || Coordinate || sin [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci19.gif align="absBottom"]] || csc [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci19.gif align="absBottom"]] || cos [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci19.gif align="absBottom"]] || sec [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci19.gif align="absBottom"]] || tan [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci19.gif align="absBottom"]] || cot [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci19.gif align="absBottom"]] ||
 * 0, 0º, [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci18.gif align="absBottom"]] || (1,0) || 0 || undefined || 1 || 1 || 0 || undefined ||
 * 90º, [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci15.gif align="absBottom"]] || (0,1) || 1 || 1 || 0 || undefined || undefined || 0 ||
 * 180º, [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci16.gif]] || (-1,0) || 0 || undefined || -1 || -1 || 0 || undefined ||
 * 270º, [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT3/unitci17.gif align="absBottom"]] || (0,-1) || -1 || -1 || 0 || undefined || undefined || 0 ||

[]
 * HELPUL WEBISTE:**


 * Examples:**
 * What is the sin at 90 degrees?
 * Using rules (sin is Y value) the unit circle shows the coordinate pair at 90 degrees to be (0,1). Therefore, the sin is 1.
 * Is cos ever undefined? If not, what makes a trig ratio undefined?
 * No, cosine is not undefined, because X values are never undefined in a unit circle. Values are only undefined because if an original value is zero and one take the inverse, the denominator will become zero. A fraction with a denominator of zero is undefined.
 * What is the trig ratio of tan at zero degrees?
 * Zero, because Y/X or sin/cos is also 0/1 or just zero.




 * PORTFOLIO QUESTIONS:**


 * OBJECTIVE 8:**


 * 1. What is the sign of tan in the fourth quadrant?**
 * 2. In what quadrants is cosine negative?**
 * 3. What trig ratio has a positive sign in quadrant 4?**


 * OBJECTIVE 9:**


 * 1. Why is cotangent at 180 degrees undefined?**
 * 2. What is the value of cos at pi/2 or 90 degrees?**
 * 3. What is the value of sin at 270 degrees?**

MOLLIE TRAUB video: (please excuse in the first slide, quadrant four should read (X, -Y) and NOT (-X, -Y))  [] OR   http://screencast-o-matic.com/watch/clffnok2S

Pictures/Sources:

http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0055/quadrants.gif [] [] [] []