SOLUTION: For the given value, determine the quadrant(s) in which the terminal side of the angle lies. tan theta = 1.700 I'm confused on exactly how you find the terminal side, regardl

Algebra.Com
Question 662885: For the given value, determine the quadrant(s) in which the terminal side of the angle lies.
tan theta = 1.700
I'm confused on exactly how you find the terminal side, regardless of whether its tan, sin, cos, etc. Can it be explained like I'm 5?
Found 2 solutions by solver91311, KMST:
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


No, I can't explain it to you like you were 5. If you need it explained like you are 5, come back and ask the question again in 10 years so I can explain it to you like you were 15.

Look at the unit circle. The of the angle between the terminal ray and the positive -axis is the -coordinate of the point of intersection between the terminal ray and the unit circle. Likewise, the is the coordinate of that point. -values are positive in the I and II quadrants, negative in the III and IV quadrants. -values are positive in I and IV quadrants and negative in II and III quadrants. Since the tangent function is the quotient of sine over cosine, the tangent function is positive in the quadrants where sine and cosine have the same sign, i.e. I and III and negative where they are opposite, II and IV.

Determining which quadrant the angle 1.7 lies is dependant on whether you are measuring the angle in degrees or in radians. I highly suspect radians, but don't like to make assumptions and since you didn't specify...

If it means degrees, then clearly it is less than 90 degrees and therefore clearly in the first quadrant. However, if it is radians, then you have to decide whether 1.7 lies in the interval , that is Quadrant II, or in the interval , that is Quadrant I.



John

My calculator said it, I believe it, that settles it
The Out Campaign: Scarlet Letter of Atheism
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
I could try to explain it for a 5-year old who reads very well, and can do basic arithmetic. If you know of any such a 5-year old, I'd love to try an explanation on them for a laugh.
In the meantime, I'll try an explanation that I believe might work for you. If it doesn't you can let me know via the website in a "thank you" note. I would answer.

EXPLANATION FOR YOU:
When you have a set of coordinates, like the one below, the axes divide it into four angles called quadrants.
The quadrants are named as the first, second, third, and fourth quadrants.
They use roman numerals I, II, II, and IV to label them.
They start at the right end of the x-axis and name them in order going counterclockwise, like this:

When we want to draw an angle on the coordinate plane we put the vertex on the origin, and the side we call the starting side along the positive x-axis.
The other side of the angle is called the terminal side.


As solver explains in his answer, the cosine of an angle is the x coordinate of the point where the terminal side crosses the unit circle.
(You could say it is the x coordinate of the point on the terminal side that is one unit away from the origin).
As solver said, the sine of the angle is the y coordinate of that same point.
(By the way, the unit circle solver included in his answer is extremely useful.
If you do not have a printout like that, get one).

As for then tangent,
, so
for the point where the terminal side crosses the unit circle.
The beauty of tangent is that you can pick any (x,y) point on the terminal side of the angle to calculate that ratio.
The ratio is the same for all points on the terminal side.
I circled point (1, 1.7) in the drawing below.
That angle's terminal side is in quadrant I, the first quadrant.
The angle theta in the graph, swept counter-clockwise from starting side to terminal side), is (or if measuring in radians).

But there are still more answers. The terminal side of an angle with the same tangent could be the ray (on the same line) that starts at the origin, but goes in the opposite direction (see it drawn in green below).
Calling the green angle , too.
That green angle's terminal side is in quadrant III, the third quadrant.
Now, theta (measured counter-clockwise, as tradition demands) could be (or if using radians).

NOTES (more advanced information; skip if confusing):
We envision the angles as turns, as space swept by a ray (like one of those blue arrows) going from the starting side to the terminal side.
We measure angles as positive if it is a counter-clockwise turn, and negative if it is a clockwise turn.
We even extend the idea of angle to include positive angles, and negative angles whose measure is very large in absolute value , as , and .
(You may think there is not much use for such large angles, but Shawn White would disagree. Not to mention that a manual could tell you to turn a knob to mean ).
The same terminal sides can be reached by going around some number of whole turns (clockwise or counter-clockwise), and then going the ,
or positions counter-clockwise.
All those angles can be represented by
for all integers.
or if measuring the angle in radians.
RELATED QUESTIONS

For the given value, determine the quadrant(s) in which the terminal side of the angle... (answered by Theo,rothauserc)
The terminal side of angle theta in standard position lies on the given line in the given (answered by jsmallt9)
For the given angle, name the quadrant in which the terminal side lies.... (answered by lwsshak3)
Determine the quadrant in which the terminal side of an angle of 6pie/5... (answered by Alan3354)
suppose theta is an angle in standard position whose terminal side lies in quadrant IV.... (answered by lwsshak3)
How do you figure out which quadrant the terminal side of each angle lies in based on if... (answered by Theo,Edwin McCravy)
Can someone please help me with this problem? / Determine the quadrant in which the... (answered by stanbon)
Determine in which quadrant the terminal ray of Theta lies if cot theta>0 and CSC Theta... (answered by lwsshak3)
Find the value of csc for angle in standard position if the point at (3, –1) lies on... (answered by solver91311)