Find the value of tan 270 degrees minus 𝜃 given the cos of 𝜃 is equal to negative four-fifths, where 𝜃 is greater than 90 degrees and less than 180 degrees.
We will begin by using the fact that 𝜃 lies between 90 and 180 degrees and the cos of 𝜃 is negative four-fifths to calculate the value of tan 𝜃. Using our CAST diagram, we see that angle 𝜃 lies in the second quadrant. We know that the sine of any angle in this quadrant is positive, whereas the cosine and tangent of any angle between 90 and 180 degrees is negative. This means that tan 𝜃 must be negative. Using our knowledge of Pythagorean triples, we see that the tan of angle 𝛼 on our diagram is three-quarters. This means that the tan of angle 𝜃 is negative three-quarters.
Let’s now consider the expression we’re given in this question, the tan of 270 degrees minus 𝜃. Using the periodicity of the tangent function, we know that the tan of 𝜃 plus or minus 180 degrees is equal to the tan of 𝜃. Since 270 degrees is equal to 180 degrees plus 90 degrees, we can rewrite our expression as the tan of 180 degrees plus 90 degrees minus 𝜃. And since tan of 180 degrees plus 𝜃 is equal to tan 𝜃, our expression is equal to the tan of 90 degrees minus 𝜃.
Recalling the cofunction identities, the tan of 90 degrees minus 𝜃 is equal to the cot of 𝜃. And using our knowledge of the reciprocal trigonometric functions, we know this is equal to one over tan 𝜃. We can now substitute our value of tan 𝜃 into the expression. We need to divide one by negative three-quarters. And since dividing by a fraction is the same as multiplying by the reciprocal of this fraction, this is the same as one multiplied by negative four-thirds. If 𝜃 lies between 90 degrees and 180 degrees such that the cos of 𝜃 is negative four-fifths, then the tan of 270 degrees minus 𝜃 is negative four-thirds.