State the quadrant in which theta lies.
sin(theta) > 0 and cos(theta) > 0
How can I determine this? Please explain.
The answer would be quadrant 1, correct?
What about sec(theta) > 0 and cot(theta) < 0?
Would that be in quadrant 3?
sec θ > 0
cot θ < 0
sec θ > 0
1 / cos θ > 0
If 1 / cos θ > 0 then also cos θ > 0
cot θ < 0
cos θ / sin θ < 0
If cos > 0 cot θ = cos θ / sin θ can be < 0 only if sin θ < 0
You must find quadrat where:
cos θ > 0 and sin θ < 0
In Quadrant IV, cos θ > 0, sin θ < 0
Actually, I don't think it would. How can this one be determined?
yes
To determine the quadrant in which theta lies based on sin(theta) and cos(theta), you need to consider the signs of sine and cosine in each quadrant. Here's how you can determine the quadrant:
1. Recall that sine represents the y-coordinate and cosine represents the x-coordinate in the unit circle.
2. If sin(theta) > 0 and cos(theta) > 0, it means that both the y-coordinate and the x-coordinate of the corresponding point on the unit circle are positive.
3. From this information, we can conclude that the point lies in the first quadrant (top right), where both the x and y coordinates are positive.
Therefore, the value of theta lies in the first quadrant.
I prefer Steve's and Bosnian's method:
if cos>0, right-hand quadrants.
if sin>0, upper-quadrants
if tan>0, 1st or 3rd quadrants.
(sec same as cos, csc same as sin, cot same as tan)
However, for checking, you can use the CAST method.
S|A
-+--
T|C
They correspond to the quadrants in which the trigonometric functions (cos, ALL, sin, tan) are positive.
http://mathonweb.com/help_ebook/html/cast.htm
just recall the definition of the trig functions, in terms of a standard triangle.
sin = y/r
cos = x/r
tan = y/x
since r is always positive, you just need to ask yourself
where are x and y both positive?
do you feel lucky? . . . well, do ya?