Find the sign of the expression if the terminal point determined by t is in the given quadrant. cos t X sec t in any given quadrant
If we examine the expression
cos(t)*sec(t)
and noting that sec(t)≡1/cos(t), it is possible to determine the sign (and the value) of the given expression for t in any quadrant.
To determine the sign of the expression cos t * sec t in a specific quadrant, we first need to understand the signs of cosine and secant in different quadrants.
Here's a quick review of the signs in each quadrant:
1. In the first quadrant (0° to 90°): Both cosine and secant are positive.
2. In the second quadrant (90° to 180°): Cosine is negative, while secant is positive.
3. In the third quadrant (180° to 270°): Both cosine and secant are negative.
4. In the fourth quadrant (270° to 360°): Cosine is positive, while secant is positive.
Now, let's analyze the expression cos t * sec t:
1. In the first quadrant (0° to 90°): Both cosine and secant are positive, so the expression is positive.
2. In the second quadrant (90° to 180°): Cosine is negative, and secant is positive, so the expression is negative.
3. In the third quadrant (180° to 270°): Both cosine and secant are negative, so the expression is positive.
4. In the fourth quadrant (270° to 360°): Cosine is positive, and secant is positive, so the expression is positive.
In summary, the sign of the expression cos t * sec t depends on the quadrant where the terminal point determined by t lies. It is positive in the first and fourth quadrants, and negative in the second quadrant.
To determine the sign of the expression cos(t) × sec(t) in any given quadrant, we need to consider the values of cosine and secant in that particular quadrant.
Here are the signs in each quadrant:
1. First Quadrant (0 to 90 degrees): Both cosine and secant are positive, so the expression cos(t) × sec(t) is positive.
2. Second Quadrant (90 to 180 degrees): Cosine is negative, while secant is positive. Therefore, the expression cos(t) × sec(t) is negative.
3. Third Quadrant (180 to 270 degrees): Both cosine and secant are negative, resulting in a positive value when multiplied together. Thus, the expression cos(t) × sec(t) is positive.
4. Fourth Quadrant (270 to 360 degrees): Cosine is positive, and secant is negative in this quadrant. Consequently, the expression cos(t) × sec(t) is negative.
In summary:
- First and Third Quadrants: cos(t) × sec(t) is positive.
- Second and Fourth Quadrants: cos(t) × sec(t) is negative.