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True or False: Each of the trigonometric functions is positive in only one quadrant.
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We begin by recalling that the four quadrants in the 𝑥𝑦-plane are as shown.
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We measure positive angles in a counterclockwise direction from the positive 𝑥-axis.
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This means that the first quadrant contains angles between zero and 90 degrees, the second quadrant between 90 and 180 degrees, and so on.
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One way of recalling whether the trigonometric functions are positive or negative in each quadrant is using the CAST acronym.
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The letter A in the first quadrant stands for all, as all three of sin 𝜃, cos 𝜃, and tan 𝜃 are positive when the angle 𝜃 lies between zero and 90 degrees.
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The sine of any angle in the second quadrant between 90 and 180 degrees is also positive.
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However, the cosine and tangent of any angle in this quadrant is negative.
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In the third quadrant, the tangent of any angle is positive, whereas the sine and cosine of any angle between 180 and 270 degrees is negative.
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Finally, in the fourth quadrant, the cosine of any angle is positive, whereas the sine and tangent of any angle is negative.
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We can therefore conclude that the sine function is positive in the first and second quadrants.
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The cosine function is positive in the first and fourth quadrants.
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And the tangent function is positive in the first and third quadrants.
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Summarizing this, we see that all three trigonometric functions are positive in two quadrants.
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This means that the initial statement is false.