SOLUTION: If the tangent of an angle is positive and the secant of the angle is negative, in which quadrant does the angle terminate? A. I B. II C. III D. IV E. Information given is not

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Question 1197786: If the tangent of an angle is positive and the secant of the angle is negative, in which quadrant does the angle terminate?
A. I
B. II
C. III
D. IV
E. Information given is not sufficient to determine.
Found 4 solutions by MathLover1, Alan3354, math_tutor2020, MathTherapy:
Answer by MathLover1(20850)   (Show Source):
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In Quadrant I, , and (All positive).
In Quadrant II, , and (Sine positive).
In Quadrant III, , and (Tangent positive).
In Quadrant IV, , and (Cosine positive).

so,
in quadrant I, III
Secant is reciprocal of cosine which is negative in the II and III quadrant, so
in quadrant II,III

answer is: C. III

Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website!
Look up the "Unit Circle"
It's very useful for questions like this.

Answer by math_tutor2020(3817)   (Show Source):
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sin = sine
cos = cosine
sec = secant
tan = tangent

tan = sin/cos
sec = 1/cos

Since tangent is positive, this means sin/cos is positive.
This further means one of the the following (pick one)

sin > 0 and cos > 0 (quadrant I)
sin < 0 and cos < 0 (quadrant III)
We also know secant is negative, which makes 1/cos negative and cosine is also negative.
Recall that
x = cos(theta)
If cosine is negative then we're to the left of the y axis in either quadrant II or quadrant III.
But we know we're not in quadrant II because of the previous info mentioned. We must be in quadrant III.

I recommend reviewing a unit circle.

Answer: C. III

Answer by MathTherapy(10555)   (Show Source):
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If the tangent of an angle is positive and the secant of the angle is negative, in which quadrant does the angle terminate?
A. I
B. II
C. III
D. IV
E. Information given is not sufficient to determine.

Acronym: ASTC, which stands for: All: ALL Trigonometric Ratios are POSITIVE in Quadrant I Students: SINE and ALL of its affiliated Trigonometric Ratios are POSITIVE in Quadrant II Take: TANGENT and ALL of its affiliated Trigonometric Ratios are POSITIVE in Quadrant III Calculus: COSINE and ALL of its affiliated Trigonometric Ratios are POSITIVE in Quadrant IV Given that TANGENT is positive, the angle can terminate in either Quadrant I or III. However, since the SECANT (affiliated with COSINE, as it's ) of the angle is NEGATIVE, the angle DEFINITELY terminates in the 3^rd Quadrant (CHOICE C.).