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True or False: If sin 𝜃 is equal to three-fifths and cos 𝜃 is less than zero, then tan 𝜃 is equal to negative three-quarters.
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We begin by sketching the CAST diagram for angles between zero and two 𝜋 radians.
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We know that in the first quadrant, the sine, cosine, and tangent of any angle are all positive.
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In the second quadrant, the sin of angle 𝜃 is positive, whereas the cos and tan of angle 𝜃 are negative.
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In the third quadrant, only the tangent function is positive, and in the fourth quadrant, only the cosine function is positive.
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In this question, we are told that sin 𝜃 is equal to three-fifths and is therefore positive.
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The cos of angle 𝜃 is negative.
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For both of these to be true, we know that 𝜃 must lie in the second quadrant.
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And this means that the tan of angle 𝜃 must be negative.
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Sketching a right triangle in the second quadrant, we see that this is a Pythagorean triple consisting of the three positive integers three, four, and five such that three squared plus four squared is equal to five squared.
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Since the tangent of any angle in a right triangle is equal to the opposite over the adjacent, the tan of angle 𝛼 is equal to three-quarters.
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We see from the diagram that 𝜃 is equal to 180 degrees minus 𝛼.
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And we recall that the tan of 180 degrees minus 𝛼 is equal to negative tan 𝛼.
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The tan of angle 𝜃 is therefore equal to negative three-quarters.
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And we can conclude that the statement is true.
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If sin 𝜃 is equal to three-fifths and cos 𝜃 is less than zero, then tan 𝜃 is equal to negative three-quarters.