Express tan(t) in terms of cos(t), where t is in Quadrant 3
Never mind, I figured it out!
tan(t) = -sqrt(1-cos^2(t))/cos(t)
In Quadrant 3, both the sine and cosine functions are negative. To express tan(t) in terms of cos(t), we can use the identity:
tan(t) = sin(t) / cos(t)
Since t is in Quadrant 3, we know that sin(t) is negative. We can express sin(t) in terms of cos(t) using the identity:
sin(t) = √(1 - cos^2(t))
Substituting this into the equation for tan(t), we have:
tan(t) = sin(t) / cos(t)
= (√(1 - cos^2(t))) / cos(t)
Therefore, tan(t) in terms of cos(t) in Quadrant 3 is (√(1 - cos^2(t))) / cos(t).
To express tan(t) in terms of cos(t), we can use the following trigonometric identity:
tan(t) = sin(t) / cos(t),
where sin(t) represents the sine function of t. To determine the value of sin(t), we need to consider that t is in Quadrant 3. In Quadrant 3, sine is negative.
Since cos(t) is positive in Quadrant 3, we can find its value using the following identity:
cos(t) = -√(1 - sin^2(t)).
Now, let's proceed with the calculation:
1. Start with the given identity: tan(t) = sin(t) / cos(t).
2. Identify the value of cos(t) in Quadrant 3:
cos(t) = -√(1 - sin^2(t)).
As mentioned earlier, t is in Quadrant 3, where sine is negative, so we will use the negative value:
cos(t) = -√(1 - sin^2(t)).
3. Substitute the value of cos(t) into the original identity:
tan(t) = sin(t) / (-√(1 - sin^2(t))).
Therefore, tan(t) in terms of cos(t), when t is in Quadrant 3, is given by:
tan(t) = sin(t) / (-√(1 - sin^2(t))).