How do you use the unit circle to evaluate inverse functions

In summary: It is a circle with a radius of 1. The x-axis goes from -1 to 1, and the y-axis goes from 0 to 1. The points on the unit circle represent the coordinates for the angles around the circle. In summary, Homework Statement The attempt at a solution for arcsin(-1/2) involves identifying the two points on the unit circle which have -1/2 as their sine value.
  • #1
algebra2
19
0

Homework Statement



a sample problem: arcsin(-1/2)


2. The attempt at a solution

do i look at the unit circle and find the y-coordinate or x-coordinate that has -1/2?

i did ASTC, and figure that it'd be in either quad 3 or quad 4; to tell you the truth i don't understand how to use the unit circle to figure this out.

i can enter it in the calculator but my calculator doesn't change the answer into a function, for instance 9pi/5 or 5pi/6 or 5pi/7
 
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  • #2
Just DRAW IT. Identify the points on the unit circle which have sine of -(1/2). There are two of these points. One is in Q3 and the other is in Q4. You read arcsine(-1/2) as "The angle whose sine is negative one-half". You are interested in identifying this angle.
 
  • #3
i don't understand what you are trying to say mr.

i have it drawn, i am looking at it. i see two quadrants that have the points but i do not see the points themselves.

can you elaborate on how to find the two points. what you said is was not very helpful.

which points do i look at? the x and y values? x coordinates? or y coordinates? be more specific because i don't comprehend you.
 
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  • #4
How do you define the unit circle what is x-coordinate and what is the y-coordinate they correspond to some trigonometry functions... You are trying to find the coordinate that corresponds to an angle that will give -1/2 when you take the sin of it... like arcsin (.5) = 30
 
  • #5
im lost. what?
 
  • #6
Any point on the unit circle corresponds to what? Tell me please
 
  • #7
the radius?
 
  • #8
There is a coordinate axis drawn with the unit circle or not? What does a point on the unit circle mean with reference to that coordinate axis? The name UNIT circle embodies the radius in it
 
  • #9
does it correspond to the angle in radians?
 
  • #11
yeah it looks similar to what i have drawn, that point is pi/3 or better known as the coordinate (1/2, (square root of 3) / 2 )
 
  • #12
Biest has asked twice about a point and you respond with a number. I think your confusion goes back to what you said in your first post:
do i look at the unit circle and find the y-coordinate or x-coordinate that has -1/2?
In order to use the unit circle to give you sine or cosine or their inverse functions you have to know that: cos(x) is the x coordinate and sin(x) is the y coordinate of a point on the unit circle. In other words each point is (cos(x), sin(x)). x is the angle (in radians it is the same as the distance around the circle's circumference from (1, 0) to (cos(x), sin(x)).

Draw a unit circle on a coordinate and then draw the horizontal line y= -1/2. It will cross the circle in two places. Since the y coordinate is negative those two points are in the third and fourth quadrants.
 
  • #13
algebra2 said:
yeah it looks similar to what i have drawn, that point is pi/3 or better known as the coordinate (1/2, (square root of 3) / 2 )

Read the post before this one and draw a line intersecting the x-axis through the point you mention here... that should give you one of the answers
 
  • #14
one of the answers is 5pi/3?
 
  • #15
30 degrees off
 
  • #16
well you see, the line intersecting the pi/3 also intersects 5pi/3...
 
  • #17
Yes, but you are looking at the x-coordinate not the y-coordinate... i wanted you to look at the lower quadrants. You still have not understood what the POINTS on the unit circle are... that is key here.
 
  • #18
ok can we start with the points then
 
  • #19
there is a link to a diagram... look at it carefully. it defines what the points are
 
  • #20
where at
 
  • #21
the link i mean.
 
  • #23
how does that define what the points are
 
  • #24
i guess you gave up on me huh
 
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  • #25
Now will you calm down... I have other things to do as well... It clearly shows what the x and y coordinates are... cos t = x and sin t = y. so if you want sin t = -.5 you might want to look in that area.
 
  • #26
symbolipoint said:
Just DRAW IT. Identify the points on the unit circle which have sine of -(1/2). There are two of these points. One is in Q3 and the other is in Q4. You read arcsine(-1/2) as "The angle whose sine is negative one-half". You are interested in identifying this angle.

I made one small conceptual mistake regarding the possible values for arcsin(). Arcsin() and sin() are inverses of each other. The interval for the angle values for arcsin() is angles measures between negative and positive pi/2.

To clarify the meaning of "unit circle", this is a circle of radius equal to 1 unit, and centered at the origin [ point (0, 0) ] on a cartesian coordinate system.
 

Related to How do you use the unit circle to evaluate inverse functions

1. What is the unit circle?

The unit circle is a circle with a radius of 1, centered at the origin (0,0) on a coordinate plane. It is used in mathematics to understand and solve trigonometric functions.

2. How do you use the unit circle to evaluate inverse functions?

To use the unit circle to evaluate inverse functions, you first need to understand the values of sine, cosine, and tangent for each angle on the unit circle. By knowing these values, you can then use them to find the inverse functions of sine, cosine, and tangent for any given angle.

3. What is the process for finding inverse functions using the unit circle?

The process for finding inverse functions using the unit circle involves first identifying the angle on the unit circle and then finding the corresponding values of sine, cosine, and tangent for that angle. These values can then be used to find the inverse functions for that angle.

4. Can the unit circle be used to evaluate all inverse functions?

No, the unit circle can only be used to evaluate the inverse functions of sine, cosine, and tangent. Other inverse functions, such as arcsine, arccosine, and arctangent, require different methods for evaluation.

5. How does understanding the unit circle help in solving trigonometric equations?

Understanding the unit circle allows you to identify angles and their corresponding values of sine, cosine, and tangent without having to use a calculator. This can be helpful when solving trigonometric equations, as you can use the unit circle to find the values needed to solve the equation, rather than relying on a calculator.

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