Question Video: Using Pythagorean Identities to Evaluate Trigonometric Expressions Mathematics

Find the value of sec (𝜃) csc (𝜃) − cot (𝜃), given that 180° < 𝜃 < 270° and sin (𝜃) = −(3/5).

04:06

Video Transcript

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.

We begin by sketching the CAST diagram as shown. Since 𝜃 is between 180 and 270 degrees, we know it lies in the third quadrant. 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. As sin 𝜃 is equal to negative three-fifths, we can sketch a right triangle in the third quadrant. 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. 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.

We can also see from the diagram that 𝜃 is equal to 180 degrees plus 𝛼. From our knowledge of related angles, we know that the cos of 180 degrees plus 𝛼 is equal to negative cos 𝛼. And the tan of 180 degrees plus 𝛼 is equal to the tan of 𝛼. This means that cos 𝜃 is equal to negative four-fifths and tan 𝜃 is equal to three-quarters. This ties in with the fact that we know that tan 𝜃 must be positive and cos 𝜃 must be negative. 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. 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.

We are now in a position to find the value of sec 𝜃 multiplied by csc 𝜃 minus cot 𝜃. This is equal to negative five-quarters multiplied by negative five-thirds minus four-thirds. Negative five-quarters multiplied by negative five-thirds is twenty-five twelfths. And since four-thirds is equivalent to sixteen twelfths, we have twenty-five twelfths minus sixteen twelfths. As the denominators are the same, we simply subtract the numerators, giving us nine twelfths. This can be simplified by dividing the numerator and denominator by three, giving us a final answer of three-quarters.

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.

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