Could someone please help me or direct me where to get help
What is the exact value of cos 60 as found on the unit circle?
and the value of cos 5pi/3
Please show me the steps so I understand
Thank you
Please help me with above post-thank you
Google shows you many examples of the unit circle, here is one of them
http://67pics.com/view2.php?q=Picture%20Of%20The%20Unit%20Circle&url=http://www.math.tamu.edu/~austin/unit_circle.png
notice the ordered pair for 60° is (1/2 , √3/2)
your definition of cosØ = x/r
so cos 60° = (1/2) / 1 = 1/2
find 5π/3 and do the same thing
I'm not suppose to copy the coordinates off the unit circle-I'm suppose to actually solve for the coordinates-like how do you get the coordinates for 60 degrees? that's where I'm confused
make yourself familar with the ratios of sides of the 30-60-90° and the 45-45-90° right-angled triangles
for the 30-60-90 angles the corresponding sides are
1 √3 and 2 , (notice that 1^2 + (√3)^2 = 2^2 )
so cos 60 = adjacent/hypotenuse = 1/2
sin 60 = opposite/hypotenuse = √3/2
tan 60 = opp/adj = √3/1 = √3
you can do the same thing for 30 and 45, and get all those special angles in the first quadrants.
Once you get into the other quadrants, the only thing that will change are the signs of the numbers.
these two pages might help for the triangles I described
http://www.themathpage.com/atrig/30-60-90-triangle.htm
http://www.themathpage.com/atrig/isosceles-right-triangle.htm
draw a circle with a raduis of one( from ((0,0) go left,right,up,down, one unit.
draw a triangle by connecting points (0,0)(1,1) and (1,0). you now have a right triangle with a hypotenuse of 1
(also radius on circle) the short side
(1,1)to(1,0) is 1/2.(Geometry TH short
leg is 1/2 hypotenuse)long leg is sq root of 3, from (0,0) to (1,0),which is
sq root of 3 times short leg.(Geometry TH).30 degrees is near (0,0) because is it opposite the shortest side 1/2. 60 degrees is near (1,1) and the hypotenuse is opp the right angle. Now the cosine is defined as adjacent over the hypotenuse. so the
adjacent angle (closet to 60, not opp or
the hyp) is 1/2 and the hyp is 1, so cosine of 60 degress is 1/2 over 1 which is 1/2. If you draw the picture it should make sense. hope this works.
Of course! I'd be happy to help you with finding the values of cosine.
To find the exact value of cos 60, we can use the unit circle. The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate system (0,0). It helps us visualize the values of trigonometric functions for different angles.
To get the value of cos 60, draw a line from the origin to the point on the unit circle that intersects the angle 60 degrees. This line will create a right triangle with the x-axis. The x-coordinate of this point on the unit circle will be the value of cos 60.
For cos 60, we know that the angle is a special angle on the unit circle. In a 45-45-90 right triangle, the sides are in the ratio 1:1:√2. By bisecting one of the 45-degree angles, we get a 30-60-90 right triangle. In this triangle, the sides are in the ratio 1:√3:2.
Since cos is the ratio of the adjacent side to the hypotenuse, in this case, it is the ratio of the x-coordinate to the radius, the value of cos 60 is 1/2.
Now let's move on to cos (5π/3). The value of cos (5π/3) can be found using the unit circle as well. Start with the angle 5π/3, which is equivalent to 300 degrees. Draw a line from the origin to the point on the unit circle that intersects this angle. The x-coordinate of this point will give us the value of cos (5π/3).
To find the exact value, we need to simplify the angle to a reference angle within the range of 0 to 360 degrees. In this case, we can subtract 2π (360 degrees) from 5π/3, which gives us π/3 or 60 degrees. Thus, we can use the same method as before to find that the value of cos (5π/3) is also 1/2.
So, to summarize:
- The exact value of cos 60 is 1/2.
- The exact value of cos (5π/3) is 1/2.
I hope this explanation helps you understand how to find the exact values of cosine using the unit circle! Let me know if there's anything else I can assist you with.