The terminal side of theta lies on a given line in the specified quadrant. Find the values of the six trigonometric functions of theta by finding a point on the line.
y= -3/2x in quadrant 4
To find a point on the line y = -3/2x in quadrant 4, we can let x = -2, since negative values of x lie in quadrant 4.
Let's substitute x = -2 into the equation y = -3/2x:
y = -3/2(-2)
y = 3
Therefore, the point on the line y = -3/2x in quadrant 4 is (-2, 3).
Now, we can use this point to find the values of the six trigonometric functions of theta. Since the terminal side of theta lies on this line, we can consider theta as the angle formed by the line and the positive x-axis.
To calculate the trigonometric functions, we need to find the values of the opposite, adjacent, and hypotenuse sides of the right triangle formed by the values of x and y.
Opposite Side (y) = 3
Adjacent Side (x) = -2
Hypotenuse (h) = sqrt(x^2 + y^2) = sqrt((-2)^2 + 3^2) = sqrt(4 + 9) = sqrt(13)
Now, we can calculate the six trigonometric functions:
1. Sin(theta) = opposite/hypotenuse = 3/sqrt(13)
2. Cos(theta) = adjacent/hypotenuse = -2/sqrt(13)
3. Tan(theta) = opposite/adjacent = 3/-2 = -3/2
4. Csc(theta) = 1/Sin(theta) = sqrt(13)/3
5. Sec(theta) = 1/Cos(theta) = -sqrt(13)/2
6. Cot(theta) = 1/Tan(theta) = -2/3
Therefore, the values of the six trigonometric functions of theta when the terminal side of theta lies on the line y = -3/2x in quadrant 4 are:
1. Sin(theta) = 3/sqrt(13)
2. Cos(theta) = -2/sqrt(13)
3. Tan(theta) = -3/2
4. Csc(theta) = sqrt(13)/3
5. Sec(theta) = -sqrt(13)/2
6. Cot(theta) = -2/3
To find a point on the given line that lies in quadrant 4, we need to choose a value for x in quadrant 4 and then substitute it into the equation to find the corresponding value of y.
In quadrant 4, x is positive and y is negative. Let's choose x = 2 as an example. Substituting this value into the equation, we can find the corresponding value of y:
y = -3/2 * x
y = -3/2 * 2
y = -3
Therefore, the point that lies on the line in quadrant 4 is (2, -3).
Now, let's find the values of the six trigonometric functions of theta using this point. To do this, we will use the definitions of trigonometric functions in terms of coordinates on the unit circle.
1) Sine (sin): sin(theta) = y / r, where r is the distance from the origin to the point defined by (x, y).
Since the point lies on the given line, r can be calculated using the distance formula: r = sqrt(x^2 + y^2).
Substituting the values, sin(theta) = -3 / sqrt(2^2 + (-3)^2) = -3 / sqrt(13).
2) Cosine (cos): cos(theta) = x / r.
Substituting the values, cos(theta) = 2 / sqrt(2^2 + (-3)^2) = 2 / sqrt(13).
3) Tangent (tan): tan(theta) = sin(theta) / cos(theta).
Substituting the values, tan(theta) = (-3 / sqrt(13)) / (2 / sqrt(13)) = -3 / 2.
4) Cosecant (csc): csc(theta) = 1 / sin(theta).
Substituting the values, csc(theta) = 1 / (-3 / sqrt(13)) = -sqrt(13) / 3.
5) Secant (sec): sec(theta) = 1 / cos(theta).
Substituting the values, sec(theta) = 1 / (2 / sqrt(13)) = sqrt(13) / 2.
6) Cotangent (cot): cot(theta) = 1 / tan(theta).
Substituting the values, cot(theta) = 1 / (-3 / 2) = -2/3.
Therefore, the values of the six trigonometric functions of theta for the given line in quadrant 4 are:
sin(theta) = -3 / sqrt(13)
cos(theta) = 2 / sqrt(13)
tan(theta) = -3 / 2
csc(theta) = -sqrt(13) / 3
sec(theta) = sqrt(13) / 2
cot(theta) = -2/3.