Question Video: Finding Equivalent Expressions Using the Cofunction Identity for Sine and Cosine Mathematics • First Year of Secondary School

Video Transcript

Which of the following is equal to negative sin 𝜃? We have been given four choices. Option (A) is cos of three 𝜋 over two plus 𝜃. Option (B) is cos of 𝜋 over two plus 𝜃. Option (C) is sin of three 𝜋 over two plus 𝜃. And option (D) is sin of 𝜋 over two plus 𝜃.

We notice that all angle measures have been given in radians. If we prefer to work with degrees, we can use the fact that 360 degrees equals two 𝜋 radians in order to convert our radians into degrees. For the purposes of this example, it may be helpful to recall that three 𝜋 over two radians is equivalent to 270 degrees and 𝜋 over two radians is equivalent to 90 degrees.

We begin by sketching a diagram where 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 𝜃 and cos 𝜃, whereas cos 𝜃 is defined by the 𝑥-coordinate and sin 𝜃 is defined by the 𝑦-coordinate. By looking at this in more detail, we will be able to define the sine and cosine of any angle. Using a geometric interpretation allows us to discover identities of trig functions. In fact, we will use our knowledge of the periodic properties and trigonometric functions and the congruency of triangles formed by angles in standard position on the unit circle in order to derive a few of the related angle trigonometric identities.

We will now work on locating the terminal side of the angle 𝜋 over two plus 𝜃. We are especially interested in the coordinate point where this angle intersects the unit circle because the 𝑥-coordinate will give us cosine of that angle and the 𝑦-coordinate will give us sine of that angle. We see that 𝜋 over two terminates on the positive side of the 𝑦-axis. And we recall that adding an angle measure rotates us in the counterclockwise direction. Therefore, we locate the terminal side of the angle 𝜋 over two plus 𝜃 in the second quadrant. Since the rotation of a geometric figure around the origin preserves its side lengths and its 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 that in terms of 𝜃, our coordinate point is negative sin 𝜃, cos 𝜃. Now, we must determine what this coordinate point will be in terms of 𝜋 over two plus 𝜃. As previously noted, the 𝑥-coordinate is cos of 𝜋 over two plus 𝜃 and the 𝑦-coordinate is sin of 𝜋 over two plus 𝜃. These coordinates of course relate back to option (B) and option (D). However, before coming to any final conclusions, let’s take a look at the angle three 𝜋 over two plus 𝜃.

The point at which the angle three 𝜋 over two plus 𝜃 terminates and intersects with the unit circle will have an 𝑥-coordinate of cos of three 𝜋 over two plus 𝜃 and a 𝑦-coordinate of sin of three 𝜋 over two plus 𝜃. We first locate three 𝜋 over two on the unit circle, which aligns with the negative portion of the 𝑦-axis. And then we make a counterclockwise turn of 𝜃 past the angle three 𝜋 over two. This angle terminates in the fourth quadrant and gives us a third right triangle that is congruent to the first two.

Next, we recall that all 𝑥-coordinates in the fourth quadrant are positive and all 𝑦-coordinates are negative. Therefore, the coordinate point is sin 𝜃, negative cos 𝜃. Let’s not forget that in terms of three 𝜋 over two plus 𝜃, the 𝑥-coordinate is cos of three 𝜋 over two plus 𝜃 and the 𝑦-coordinate is sin of three 𝜋 over two plus 𝜃.

Now that we have completed our geometric reasoning, we have found an equivalent expression in terms of 𝜃 for each of the four possible answers. We found that cos of three 𝜋 over two plus 𝜃 equals sin of 𝜃 and that cos of 𝜋 over two plus 𝜃 equals negative sin 𝜃. sin of three 𝜋 over two plus 𝜃 equals negative cos 𝜃, and finally, sin of 𝜋 over two plus 𝜃 equals cos 𝜃. In conclusion, since only one of these options equals negative sin of 𝜃, our final answer is cos of 𝜋 over two plus 𝜃.