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Question Video: Simplifying Trigonometric Expressions Using Related Angle Trigonometric Identities Mathematics

Simplify sin (360Β° βˆ’ πœƒ).

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Video Transcript

Simplify sin of 360 degrees minus πœƒ.

To do this, let’s consider the unit circle. We’ll let πœƒ be an angle in standard position. And the point π‘₯, 𝑦 will be the coordinates of the point of intersection between the unit circle and the terminal side of πœƒ. The sin of πœƒ will equal 𝑦 over one. Therefore, we can label this distance as sin of πœƒ. Now we want to know where 360 degrees minus πœƒ would fall on our unit circle. The angle 360 degrees minus πœƒ in a counterclockwise direction would look like this. To calculate the sin of 360 degrees minus πœƒ, we would use its reference angle.

Here, we have a reflection in the π‘₯-axis. Recall that we’ve already said 𝑦 equals the sin of πœƒ. Therefore, negative 𝑦 equals the negative sin of πœƒ, which is also equal to the sin of 360 degrees minus πœƒ. This illustrates a principle that we know about the related angle identities for trigonometric functions. And that is that the sin of 360 degrees minus πœƒ is equal to the negative sin of πœƒ. We’ve been able to simplify sin of 360 degrees minus πœƒ and show that it’s equal to negative sin of πœƒ.

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