Find cosq and sinq if the terminal side of q lies along the line y = - 2 x in
quadrant IV.
take x = 1
then y = -2
hypotenuse = sqrt (1+4) = sqrt 5
then
sin q = -2/sqrt 5
cos q = 1/sqrt 5
To find cos(q) and sin(q) for an angle q, we need to determine the values of the cosine and sine functions at that angle.
Given that the terminal side of the angle q lies along the line y = -2x in quadrant IV, we can visualize it as follows:
| . (x, y)
| .
| .
| .
| .
| .
|.
---------------------------------
In quadrant IV, x is positive, and y is negative. Therefore, we can determine the coordinates of the point (x, y) on the terminal side of q as follows:
Since the equation of the line is y = -2x, we substitute x into the equation to find y:
y = -2x
We know that x and y are coordinates, so we can set x = r * cos(q) and y = r * sin(q), where r is the length of the line segment from the origin to the point (x, y) on the terminal side of q.
Substituting these values into the equation y = -2x, we have:
r * sin(q) = -2 * (r * cos(q))
Since r cannot be zero, we can divide both sides of the equation by r:
sin(q) = -2 * cos(q)
Now, we can rewrite the equation cos(q) = sin(q) / (-2).
Using the trigonometric identity tan(q) = sin(q) / cos(q), we can substitute this value into the equation:
cos(q) = tan(q) / (-2)
Now that we have an equation relating cos(q) to tan(q), we can use a scientific calculator or a table of trigonometric values to find the value of cos(q).
Similarly, we can use the equation sin(q) = -2 * cos(q) to find the value of sin(q).
Note: If you're using a scientific calculator, make sure it is set to the correct angle unit (degrees or radians) for your calculation.