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.
