Determine the quadrant in which each lies:
sec0<0,cot0<0
sin0<0,tan0>0
Use "All Students Take Calculus" to find which of the functions are positive in which quadrant. The positive quadrants are for:
S|A
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T|C
To find which quadrant will have sec<0 and cot<0, you look up the quadrant where cos<0 and tan<0.
For cos<0, it will be either quad.II or quad.III.
For tan<0, it will be either quad.II or quad.IV.
So quadrant II will satisfy both conditions.
You can try the next one. Post if you have problems.
To determine the quadrant in which each expression lies, we need to reference the trigonometric ratios and their signs in each quadrant.
1. sec(θ) = 1/cos(θ), where θ is the angle.
Given: sec(θ) < 0, cot(θ) < 0
Since sec(θ) is negative, it means that cos(θ) must be negative. In which quadrants is cos(θ) negative? Cosine is negative in the 2nd and 3rd quadrants.
Next, since cot(θ) is negative, it means that tan(θ) must also be negative. In which quadrants is tan(θ) negative? Tangent is negative in the 2nd and 4th quadrants.
Therefore, the angle θ lies in the 2nd quadrant.
2. sin(θ) < 0, tan(θ) > 0
Since sin(θ) is negative, it means that θ must be in the 3rd or 4th quadrants, as sine is negative in these two quadrants.
Next, since tan(θ) is positive, it means that θ must be in the 1st or 3rd quadrants, as tangent is positive in these two quadrants.
Therefore, the angle θ lies in the 3rd quadrant.
To summarize:
1. The angle θ in sec(θ) < 0, cot(θ) < 0 lies in the 2nd quadrant.
2. The angle θ in sin(θ) < 0, tan(θ) > 0 lies in the 3rd quadrant.