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Find the value of sec 𝜃 multiplied by csc 𝜃 minus cot 𝜃, given that 𝜃 is greater than 180 degrees and less than 270 degrees and sin 𝜃 is equal to negative three-fifths.
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We begin by sketching the CAST diagram as shown.
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Since 𝜃 is between 180 and 270 degrees, we know it lies in the third quadrant.
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We know that for any angle in this quadrant, the tangent and cotangent of the angle are positive, whereas sin 𝜃, cos 𝜃, csc 𝜃, and sec 𝜃 are all negative.
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As sin 𝜃 is equal to negative three-fifths, we can sketch a right triangle in the third quadrant.
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This right triangle is a Pythagorean triple consisting of three positive integers three, four, and five such that three squared plus four squared is equal to five squared.
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Since cos 𝛼 is equal to the adjacent over the hypotenuse and tan 𝛼 is equal to the opposite over the adjacent, from our diagram, we have cos 𝛼 is equal to four-fifths and tan 𝛼 is equal to three-quarters.
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We can also see from the diagram that 𝜃 is equal to 180 degrees plus 𝛼.
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From our knowledge of related angles, we know that the cos of 180 degrees plus 𝛼 is equal to negative cos 𝛼.
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And the tan of 180 degrees plus 𝛼 is equal to the tan of 𝛼.
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This means that cos 𝜃 is equal to negative four-fifths and tan 𝜃 is equal to three-quarters.
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This ties in with the fact that we know that tan 𝜃 must be positive and cos 𝜃 must be negative.
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Next, the reciprocal trigonometric identities tell us that csc 𝜃 is equal to one over sin 𝜃, sec 𝜃 is equal to one over cos 𝜃, and cot 𝜃 is equal to one over tan 𝜃.
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This means that csc 𝜃 is equal to negative five-thirds, sec 𝜃 is equal to negative five-quarters, and cot 𝜃 is equal to four-thirds.
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We are now in a position to find the value of sec 𝜃 multiplied by csc 𝜃 minus cot 𝜃.
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This is equal to negative five-quarters multiplied by negative five-thirds minus four-thirds.
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Negative five-quarters multiplied by negative five-thirds is twenty-five twelfths.
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And since four-thirds is equivalent to sixteen twelfths, we have twenty-five twelfths minus sixteen twelfths.
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As the denominators are the same, we simply subtract the numerators, giving us nine twelfths.
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This can be simplified by dividing the numerator and denominator by three, giving us a final answer of three-quarters.
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If 𝜃 lies between 180 and 270 degrees and sin 𝜃 is equal to negative three-fifths, then sec 𝜃 multiplied by csc 𝜃 minus cot 𝜃 is equal to three-quarters.