Video Transcript
Find cot of 180 degrees plus 𝜃 given 𝜃 is in standard position and its terminal side passes through the point negative 21 over 29, negative 20 over 29.
We will begin by sketching the unit circle and marking on angle 𝜃 in standard position. Any angle in standard position is measured from the positive 𝑥-axis. And if the angle is positive, we measure in the counterclockwise direction. We are told that the terminal side of angle 𝜃 passes through the point where both the 𝑥- and 𝑦-coordinates are negative. This means that this point lies in the third quadrant as shown. Any point 𝑃 that lies on the unit circle has coordinates cos 𝜃, sin 𝜃. This means that cos 𝜃 is equal to negative 21 over 29 and sin 𝜃 is equal to negative 20 over 29.
In this question, we are asked to find the value of the cot of 180 degrees plus 𝜃. Since the cotangent function is the reciprocal of the tangent function, then cot 𝜃 is equal to one over tan 𝜃. We also know that the tan of any angle 𝜃 is equal to sin 𝜃 over cos 𝜃. And since we know the values of sin 𝜃 and cos 𝜃, tan 𝜃 is equal to negative 20 over 29 divided by negative 21 over 29. Dividing by a fraction is the same as multiplying by the reciprocal of that fraction. We can then cancel a factor of 29 from the numerator and denominator, leaving us with negative 20 over one multiplied by negative one over 21. As multiplying two negatives gives a positive, the tan of angle 𝜃 is equal to 20 over 21.
We could use this value to find the cot of angle 𝜃. However, we will begin by trying to calculate the tan of 180 degrees plus 𝜃. The point 𝑄 on our diagram has coordinates cos 180 degrees plus 𝜃, sin 180 degrees plus 𝜃. And as this point lies in the first quadrant, we know that the 𝑥- and 𝑦-coordinates will both be positive. Using the symmetry of the unit circle, the 𝑥-coordinate, the cos of 180 degrees plus 𝜃, must be equal to 21 over 29. Likewise, the sin of 180 degrees plus 𝜃 is equal to 20 over 29.
Using the fact that tan 𝜃 is equal to sin 𝜃 over cos 𝜃, the tan of 180 degrees plus 𝜃 is equal to 20 over 21. This is equal to the tan of 𝜃, and this identity holds for any angle 𝜃. Using the reciprocal identity, this means that the cot of 180 degrees plus 𝜃 must be equal to the cot of 𝜃. And since this is equal to one divided by 20 over 21, we can conclude that the cot of 𝜃 is equal to 21 over 20 and the cot of 180 degrees plus 𝜃 is also equal to 21 over 20.
