The terminal side of angle alpha is in the fourth quadrant, and sin alpha=-1/2, find the values of the other functions.

From the ratios of sides of the 30-60-90 triangle you know that if

sinØ= +1/2 , then Ø = 30°

Also in the 4th quadrant, both the sine and tangent and of course their reciprocals are negative, the cosine is positive.

so

sinØ = -1/2 csc‚ = -2
cosØ = √3/2 secØ = 2/√3
tanØ = - 1/√3 cotØ = -√3

To find the values of the other trigonometric functions, we need to determine the values of cosine, tangent, secant, cosecant, and cotangent for the given angle.

Given that the terminal side of angle alpha is in the fourth quadrant and sin alpha = -1/2, we can determine the following:

1. Sine (sin alpha) = -1/2
Since sine is negative in the fourth quadrant, and sin alpha = -1/2, we can deduce that the value of sin alpha is -1/2.

2. Cosine (cos alpha):
In the fourth quadrant, cosine is positive. We can use the Pythagorean identity, sin^2 alpha + cos^2 alpha = 1, to find the value of cosine.
(-1/2)^2 + cos^2 alpha = 1
1/4 + cos^2 alpha = 1
cos^2 alpha = 1 - 1/4
cos^2 alpha = 3/4
Taking the square root of both sides (remembering that cosine is positive in the fourth quadrant), we get:
cos alpha = sqrt(3/4)
cos alpha = sqrt(3)/2

3. Tangent (tan alpha):
Tangent is calculated as the ratio of sine to cosine: tan alpha = sin alpha / cos alpha.
tan alpha = (-1/2) / (sqrt(3)/2)
tan alpha = -1 / sqrt(3)
To rationalize the denominator, we multiply both numerator and denominator by sqrt(3):
tan alpha = (-1 / sqrt(3)) * (sqrt(3) / sqrt(3))
tan alpha = -sqrt(3) / 3

4. Secant (sec alpha):
Secant is the reciprocal of cosine: sec alpha = 1 / cos alpha.
sec alpha = 1 / (sqrt(3)/2)
sec alpha = 2 / sqrt(3)
To rationalize the denominator, we multiply both numerator and denominator by sqrt(3):
sec alpha = (2 / sqrt(3)) * (sqrt(3) / sqrt(3))
sec alpha = 2 * sqrt(3) / 3

5. Cosecant (csc alpha):
Cosecant is the reciprocal of sine: csc alpha = 1 / sin alpha.
csc alpha = 1 / (-1/2)
csc alpha = -2

6. Cotangent (cot alpha):
Cotangent is the reciprocal of tangent: cot alpha = 1 / tan alpha.
cot alpha = 1 / (-sqrt(3) / 3)
cot alpha = -3 / sqrt(3)
To rationalize the denominator, we multiply both the numerator and denominator by sqrt(3):
cot alpha = (-3 / sqrt(3)) * (sqrt(3) / sqrt(3))
cot alpha = -3 * sqrt(3) / 3
cot alpha = -sqrt(3)

So, the values of the other trigonometric functions for an angle alpha in the fourth quadrant, with sin alpha = -1/2, are:
cos alpha = sqrt(3)/2
tan alpha = -sqrt(3) / 3
sec alpha = 2 * sqrt(3) / 3
csc alpha = -2
cot alpha = -sqrt(3)

To find the values of the other trigonometric functions with the given information, we need to use the reference angle.

1. Start by determining the reference angle:
Since the terminal side of angle alpha is in the fourth quadrant, we need to find the angle in the first quadrant that has the same sine value as angle alpha. In the first quadrant, the sine is always positive. So, to find the reference angle, we take the absolute value of the sine:
reference angle = |sin alpha| = |-1/2| = 1/2.

2. Determine the values of the other trigonometric functions using the reference angle:
a. Sine (sin):
Since the sine is negative in the fourth quadrant, the sine of angle alpha is:
sin alpha = -1/2.

b. Cosine (cos):
In the first quadrant, the cosine is positive. Therefore, the cosine of angle alpha is the same as the cosine of the reference angle:
cos alpha = cos(reference angle) = cos(1/2).

c. Tangent (tan):
The tangent is the ratio of the sine to the cosine:
tan alpha = sin alpha / cos alpha = (-1/2) / cos(1/2).

d. Secant (sec):
The secant is the reciprocal of the cosine:
sec alpha = 1 / cos alpha = 1 / cos(1/2).

e. Cosecant (csc):
The cosecant is the reciprocal of the sine:
csc alpha = 1 / sin alpha = 1 / (-1/2).

f. Cotangent (cot):
The cotangent is the reciprocal of the tangent:
cot alpha = 1 / tan alpha = 1 / [(-1/2) / cos(1/2)].

Use a calculator to determine the approximate values of the trigonometric functions by plugging in the reference angle (1/2) into the corresponding trigonometric function.