cos x -1/2 x lies in the third quadrant find the other five trigonometery functions
cos x - 1 / 2 x
can't lies in the third quadrant.
Go on:
wolframalpha . c o m
an type cos x - 1 / 2 x
you will see graph
maybe you mean cos x = -1/2
Since cosθ = x/r, and θ is in QIII
y = -√3
sinθ = y/r = -√3/2
tanθ = y/x = √3
cotθ = x/y = 1/√3
secθ = r/x = -2
cscθ = r/y = -2/√3
To find the other five trigonometric functions (sin(x), tan(x), sec(x), csc(x), and cot(x)) of an angle given the value of one trigonometric function, you can use the following steps:
1. Start by sketching the angle in the third quadrant. In this case, the angle lies in the third quadrant, which means it has a negative cosine value (cos(x) < 0) and a negative sine value (sin(x) < 0).
2. Since the value of cos(x) is given as -1/2, you can determine the adjacent side and hypotenuse of a right triangle formed by the angle x. Since cosine is equal to the ratio of the adjacent side to the hypotenuse, you can set the adjacent side as 1 and the hypotenuse as 2 (taking the absolute values since they are lengths).
3. Use the Pythagorean theorem to find the length of the opposite side. Since the adjacent side is 1 and the hypotenuse is 2, you can calculate the opposite side length as follows:
opposite side² = hypotenuse² - adjacent side²
opposite side² = 2² - 1²
opposite side² = 4 - 1
opposite side² = 3
opposite side = √3
4. Now that you have all three sides of the right triangle (adjacent = 1, opposite = √3, hypotenuse = 2), you can determine the values of the other trigonometric functions:
- sine (sin(x)) = opposite/hypotenuse = √3/2 (since sin(x) < 0)
- tangent (tan(x)) = opposite/adjacent = √3/1 = √3 (since tan(x) < 0)
- cosecant (csc(x)) = 1/sine(x) = 2/√3 = 2√3/3 (reciprocal of sin(x), csc(x) < 0)
- secant (sec(x)) = 1/cos(x) = 1/(-1/2) = -2 (reciprocal of cos(x), sec(x) < 0)
- cotangent (cot(x)) = 1/tan(x) = 1/(√3) = √3/3 (reciprocal of tan(x), cot(x) < 0)
Therefore, in the third quadrant, when cos(x) = -1/2, the other trigonometric functions are:
sin(x) = -√3/2
tan(x) = -√3
csc(x) = -2√3/3
sec(x) = -2
cot(x) = √3/3