State the quadrant in which θ lies.
1.) CSC θ is greater than 0 and TAN θ is
less than 0.
2.) SEC θ is greater than 0 and SIN θ is
less than 0.
if cscØ > 0 , then sinØ > 0
the sine is + in I and II , by the CAST rule
the tangent is - in II and IV , by the CAST rule
so Ø in in II
so the 2nd question the same way. ( IV )
1.) For the first condition, CSC θ is greater than 0. Since CSC (cosecant) is equal to 1/SIN (sine), this means that SIN θ is also greater than 0. Additionally, TAN θ is less than 0.
In the coordinate plane, SIN θ represents the y-coordinate and TAN θ represents the ratio of the y-coordinate to the x-coordinate. If SIN θ is greater than 0 and TAN θ is less than 0, this means that the y-coordinate is positive (greater than 0) and the x-coordinate is negative (less than 0).
Based on this information, we can determine that θ lies in the second quadrant.
2.) For the second condition, SEC θ is greater than 0. Since SEC (secant) is equal to 1/COS (cosine), this means that COS θ is greater than 0. Additionally, SIN θ is less than 0.
In the coordinate plane, COS θ represents the x-coordinate and SIN θ represents the y-coordinate. If COS θ is greater than 0 and SIN θ is less than 0, this means that the x-coordinate is positive (greater than 0) and the y-coordinate is negative (less than 0).
Based on this information, we can determine that θ lies in the fourth quadrant.
To determine the quadrant in which θ lies based on the given trigonometric equations, we can use the knowledge of the signs of trigonometric functions in different quadrants:
1.) For CSC θ > 0 and TAN θ < 0:
- CSC θ represents the cosecant function, and if it is positive, it means that sin θ is also positive.
- TAN θ represents the tangent function, and if it is negative, it means that tan θ is also negative.
Based on these conditions:
- Since CSC θ is positive, sin θ is positive. In which quadrants is sin positive? The answer is Quadrant I and Quadrant II.
- Since TAN θ is negative, tan θ is negative. In which quadrants is tan negative? The answer is Quadrant II and Quadrant IV.
Therefore, the only quadrant where both conditions are met is Quadrant II.
2.) For SEC θ > 0 and SIN θ < 0:
- SEC θ represents the secant function, and if it is positive, it means that cos θ is also positive.
- SIN θ represents the sine function, and if it is negative, it means that sin θ is negative.
Based on these conditions:
- Since SEC θ is positive, cos θ is positive. In which quadrants is cos positive? The answer is Quadrant I and Quadrant IV.
- Since SIN θ is negative, sin θ is negative. In which quadrants is sin negative? The answer is Quadrant III and Quadrant IV.
Therefore, the only quadrant where both conditions are met is Quadrant IV.
In summary:
1.) For CSC θ > 0 and TAN θ < 0, the angle θ lies in Quadrant II.
2.) For SEC θ > 0 and SIN θ < 0, the angle θ lies in Quadrant IV.