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 the values of the six trigonometric functions of theta, we first need to find a point on the line in the specified quadrant. In this case, the line is y = -3/2x and the specified quadrant is quadrant 4.
In quadrant 4, the x-coordinate is positive, and the y-coordinate is negative. We can choose any value for x as long as it is positive, and then calculate the corresponding y-coordinate using the equation of the line.
Let's pick an x-value of 2. Substitute this value into the equation y = -3/2x:
y = -3/2 * 2
y = -3
So a point on the line in quadrant 4 is (2, -3).
Now, let's calculate the values of the trigonometric functions using this point:
1. The sine of theta (sin(theta)) is the ratio of the y-coordinate to the hypotenuse. In this case, the hypotenuse is the distance from the origin to the point (2, -3), which can be found using the Pythagorean theorem:
hypotenuse = sqrt((2^2) + (-3^2))
= sqrt(4 + 9)
= sqrt(13)
Therefore, sin(theta) = y-coordinate / hypotenuse = -3 / sqrt(13).
2. The cosine of theta (cos(theta)) is the ratio of the x-coordinate to the hypotenuse. So cos(theta) = x-coordinate / hypotenuse = 2 / sqrt(13).
3. The tangent of theta (tan(theta)) is the ratio of the y-coordinate to the x-coordinate. So tan(theta) = y-coordinate / x-coordinate = -3 / 2.
4. The cosecant of theta (csc(theta)) is the reciprocal of the sine of theta. Therefore, csc(theta) = 1 / sin(theta) = sqrt(13) / (-3).
5. The secant of theta (sec(theta)) is the reciprocal of the cosine of theta. So sec(theta) = 1 / cos(theta) = sqrt(13) / 2.
6. The cotangent of theta (cot(theta)) is the reciprocal of the tangent of theta. Hence, cot(theta) = 1 / tan(theta) = 2 / (-3).
To summarize:
- 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)