Find every angle (theta) between 0 & 360 degrees for which the ratio of sin theta and cosine theta is -3.00
tanTheta=-3
theta=arctan(-3)
Now tan theta is negative in the second and fourth quadrant.
put this in your google search window: arctan(-3)=
and you get -71.57 deg (appx).That is the fourth quadrant, for the second quadrant, add 180 degrees.
To find every angle between 0 and 360 degrees for which the ratio of sin(theta) and cos(theta) is -3.00, we can use the following steps:
1. Start with the given ratio, sin(theta)/cos(theta) = -3.00.
2. Recall that sin(theta) = opposite/hypotenuse and cos(theta) = adjacent/hypotenuse in a right triangle.
3. Rewrite the ratio using the definitions of sine and cosine:
opposite/hypotenuse / adjacent/hypotenuse = -3.00.
4. Simplify the ratio by canceling out the common factor of hypotenuse:
(opposite/adjacent) = -3.00.
5. Now, consider the geometric interpretation of the ratio -3.00 = (opposite/adjacent). We know that for any angle theta, the sine function can be negative or positive, but cosine is always positive for angles between 0 and 360 degrees.
6. Based on this information, we can conclude that the ratio -3.00 corresponds to an angle theta in either the second quadrant (where sin(theta) is positive and cos(theta) is negative) or the fourth quadrant (where sin(theta) is negative and cos(theta) is positive).
7. To find the exact values of theta, we can use the inverse trigonometric functions. In this case, theta can be found using the arcsin(-3.00) or arccos(-3.00).
8. Calculate the angles using a calculator or software. Note that the angles obtained will be in radians, so convert them to degrees, if necessary.
9. The angles obtained will be the solutions to the equation sin(theta)/cos(theta) = -3.00 between 0 and 360 degrees.