if cos(x)= 1/2, then how do you find x?

Answer 1

cos(x)= 1/2
arccos(cos(x))=arccos(1/2)
x=arccos(1/2)
x=60 (degrees) or pi/3 (radians) dependinhg on your level of math

Answer 2

One uses a calculator, a computer spreadsheet or a pre-printed table.

There are a few functions to find the answer (one is an improper integral, another is an infinite series).

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The inverse of the function cosine is called arccosine, with symbol arccos (pronounced "arc cos" in two syllables)

if cos(x) = 1/2, then what you seek is
x = arccos(1/2)

The infinite series for arccos(z) is

(pi/2) - { z + (1/2)(z^3/3) + (3/8)(z^5/5) + (15/48)(z^7/7) + ... + (An/Bn)(z^[2n-1]/(2n-1)}

(This will give x in radians)

The expression inside the braces is arcsin(z)

for the nth term inside the brace:
An = the product of the first nth odd integers (1 is included)
Bn = the product of the first nth even integers

example, for the 6th term (n = 6)

An = 1*3*5*7*9*11 = 10,395
Bn = 2*4*6*8*10*12 = 46,080
z^[2n-1] = z^11

6th term is:

(10,395/46,080)(z^11/11)

If z is small, then by the time you are at this stage, you are only changing the number by very little.

z = 1/2 then the 6th term =
10,395 / ( 46,080 * 11 * 2^11) = 0.00001

If all you need is 4 decimals, then you'd stop here.

There are other series, some of them converge faster.

Answer 3

First... forget degrees unless you're in very elementary geometry. In trig you'll be using radians. Granted, they're not as neat, but they're scalar numbers, not units of measurements. There's nothing you can do with 5 degrees but find it on a compass or map or sextant. But you can't add 2 to 5 degrees. However, with radians, you can add numbers, multiply by numbers, etc.

Here are the values you should know. You can memorize them if you want. Whether you choose to do so or not, you will become nauseatingly familiar with them over the course of years of math. These and Pythagorean triples are almost always included on standardized tests.

Functions on the
Unit Circle
θ . . . . sinθ . . . . cosθ
0 . . . √(0/4). . . .√(4/4)
π/6 . √(1/4). . . . √(3/4)
π/4 . √(2/4). . . . √(2/4)
π/3 . √(3/4). . . . √(1/4)
π/2 . √(4/4). . . . √(0/4)

In your question, you want cos(x)=1/2
1/2 = √(1/4) which is cos(π/3 ), so x = π/3..... sort of.
You see, the sine and cosine are cyclic functions.. that means that they keep repeating themselves, going throug the same range of values over and over and over....

It turns out that cos(-π/3)... or cos(5π/3 )... also = 1/2 and every time you add 2π to either of those values, you get the same values for the cosine. So, for any integer n, x = π/3 + 2nπ, or x = 5π/3 + 2nπ.

Not as neat as algebra when you only got one answer... is it? ;-)

Answer 4

remember the unit circle that your dear teacher used to illustrate the circular [trigonometric] functions. the unit circle will remind you the definition of the cosine of the angle: the ratio of the length of the adjacent side to the hypotenuse.

If cos (x)=1/2, x is an angle in a right triangle with leg and hypotenuse that have a ratio of 1:2. That's a special right triangle: 30, 60, 90 triangle. The angle is turns out to be 60 deg. [The leg adjacent to the angle x is the shortest of the three side of the triangle, so x is not the 30 degree angle.]

Answer 5

This involves knowledge/memorization of the unit circle coordinates. Remember that cosine represents the x-coordinate for the unit circle.

Cosine is equal to positive 1/2, and cosine is positive in quadrants 1 and 4. Therefore, you should have a solution in those quadrants.

Assuming an interval restriction of [0, 2pi), the answer is

x = {pi/3, 5pi/3}

Answer 6

cos(x)= 1/2 => x must be in the first or the fourth quadrant.

Since the reference angle is pi/3, the general solution is
x = pi/3 + 2n*pi
or
x = -pi/3 + 2n*pi, where n can be any integer.

Answer 7

60 degrees (it's a 30/60/90 triangle)

P.S. I would recommend having the 45/45/90 and 30/60/90 triangles memorized for any tests you will be having...

Answer 8

cos x = 1/2
x = 60° , x = 300°
or
x = π / 3 , x = 5π / 3

Answer 9

cos(x)= 1/2 = 0.50
x = 60º

press shift cos 0.5 =