Video Transcript
Simplify tan of 90 degrees plus 𝜃.
We recall that if the angle 𝜃 is in standard position on a set of axes with a unit circle centered at the origin, then we can use the coordinates of the intersection between the terminal side of the angle and the unit circle to define sin of 𝜃 and cos of 𝜃. By looking at this in more detail, we will be able to define the sine and cosine of any angle. Since tan 𝜃 is equal to sin 𝜃 over cos 𝜃, we can use these values to evaluate the tangent of any angle or any reciprocal trigonometric function. This geometric interpretation allows us to discover identities of the trig functions.
In fact, it is possible to memorize some correlated trigonometric identities, such as sin of 90 plus 𝜃, cos of 90 plus 𝜃, and even tan of 90 plus 𝜃. Let’s say that we don’t have these trig identities memorized yet. In this case, we can derive these identities by using the periodic properties of trig functions and the congruency of triangles formed by angles in standard position on the unit circle.
We will begin by locating the terminal side of the angle 90 degrees plus 𝜃 on the unit circle. To do this, we recall that the terminal side of 90 degrees is found on the positive side of the 𝑦-axis and that rotating counterclockwise around the origin represents adding an angle measure, whereas rotating clockwise represents subtracting an angle measure. We have sketched the terminal side of the angle 90 degrees plus 𝜃 in the second quadrant. Since the rotation of a geometric figure around the origin preserves its side lengths and angle measures, we now have a pair of congruent right triangles. These congruent triangles have in common a green hypotenuse of length one and a shorter orange side of length sin 𝜃 and a longer pink side of length cos 𝜃.
Now, within the second quadrant, all 𝑥-coordinates are negative and all 𝑦-coordinates are positive. Therefore, we assign a negative value to the length of the orange side and a positive value to the length of the pink side. This means our coordinate point is negative sin 𝜃, cos 𝜃.
We should note that these coordinate points are written in terms of our original angle 𝜃. In terms of the angle 90 degrees plus 𝜃, the 𝑥-coordinate is cos of 90 degrees plus 𝜃 and the 𝑦-coordinate is sin of 90 degrees plus 𝜃. By using this geometric interpretation, we have just found the first two correlated trig identities.
We now know that sin of 90 degrees plus 𝜃 equals cos of 𝜃 and cos of 90 degrees plus 𝜃 equals negative sin 𝜃. We could now use the definition of tan 𝜃 along with these first two correlated trig identities to simplify tan of 90 degrees plus 𝜃. Using the definition of tangent in terms of sine and cosine gives us tan of 90 degrees plus 𝜃 equals sin of 90 degrees plus 𝜃 over cos of 90 degrees plus 𝜃. Then, substituting the simplified expressions found in our first two correlated trig identities gives us cos of 𝜃 over negative sin of 𝜃.
Finally, we recall that the cotangent function is the reciprocal of the tangent function. And using the fact that a positive divided by a negative is a negative, our final answer is negative cotan of 𝜃. In conclusion, the simplification of tan of 90 degrees plus 𝜃 equals negative cotan of 𝜃.
