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
let x=2, then y = (-3/2)(2) = -3
then r^2 = 2^2 + (-3)^2 = 13
r = √13
sinØ = -3/√13 , cscØ = -√13/3
cosØ = 2/√13 , secØ = √13/2
tanØ = -3/2 , cotØ = -2/3
If secθ=√13/3 and the terminal side of θ lies in quadrant IV, find sinθ.
To find a point on the given line in quadrant 4, we need to consider the signs of the coordinates. In quadrant 4, both the x and y coordinates are positive.
Let's start by finding the x-coordinate. We can set y = 0 (since the line lies on the x-axis) and solve for x:
0 = -3/2x
To solve for x, we can multiply both sides by -2/3:
0 * (-2/3) = -3/2x * (-2/3)
0 = x
So, the x-coordinate of the point on the line is 0.
Next, we can find the y-coordinate. Since the line equation is y = -3/2x, we substitute x = 0 into the equation:
y = -3/2 * 0
y = 0
So, the y-coordinate of the point on the line is also 0.
Now, we have the point on the line: (0, 0).
To find the values of the six trigonometric functions of theta, we need to consider the ratios of the sides of a right triangle formed by drawing a line from the origin (0,0) to the point on the line.
In this case, the triangle with the point (0,0) forms a right triangle with the x-axis and y-axis.
The six trigonometric functions can be defined as follows:
1. Sine (sin): sin(theta) = opposite/hypotenuse = y-coordinate/1 = 0/1 = 0
2. Cosine (cos): cos(theta) = adjacent/hypotenuse = x-coordinate/1 = 0/1 = 0
3. Tangent (tan): tan(theta) = opposite/adjacent = y-coordinate/x-coordinate = 0/0 (undefined since division by zero is undefined)
4. Cosecant (csc): csc(theta) = 1/sin(theta) = 1/0 (undefined since division by zero is undefined)
5. Secant (sec): sec(theta) = 1/cos(theta) = 1/0 (undefined since division by zero is undefined)
6. Cotangent (cot): cot(theta) = 1/tan(theta) = 1/0 (undefined since division by zero is undefined)
So, the values of the six trigonometric functions of theta are as follows:
sin(theta) = 0
cos(theta) = 0
tan(theta) = undefined
csc(theta) = undefined
sec(theta) = undefined
cot(theta) = undefined