Use 105° = 135° - 30° to find the exact value of 105°.

Question

Accepted Answer

Hello, everyone. We are asked to find the exact value of the sign of degrees. Using the relationship given below, we are told 105 degrees equals 150 degrees minus 45 degrees. We have four answer choices. First recall that if we are thinking about identities, we know that the sign of A minus B equals the sign of angle A multiplied by the cosine of angle B minus the cosine of angle. A multiplied by the sign of angle B. How this helps us is we can say that the sign of 105 degrees equals the sign of 150 degrees minus 45 degrees from there. We can use this identity. So the sign of 150 degrees minus 45 degrees is the equivalent of saying the sign of 150 degrees times the cosine of 45 degrees minus the cosine of 100 50 degrees multiplied by the sine of 45 degrees from there. Using a unit circle. We can substitute in values for these signs and cosigns the sign of 150 degrees is one half that'll be multiplied by the cosine of 45 degrees, which is radical two divided by two. And then we subtract the cosine of degrees which is negative radical three divided by two multiplied by the sine of 45 degrees which is radical two divided by two. Using our multiplication rules multiplying my first numbers together one multiplied by radical two is radical two, two multiplied by two is four in the next set. Negative radical three multiplied by radical two is negative radical six and two multiplied by two is four double negatives are like adding. So this is saying radical two divided by four plus radical six divided by four. And all of our answer choices make this into one fraction. So we are going to do that. So it's radical two plus radical six divided by four. And if I were to write that, so it was laying down, I would put the radical two plus radical six in parentheses and then divide by four. So this is our exact value and it appears to match answer choice. C so in parentheses, radical two plus radical six and then that whole set of parentheses divided by four. And that's answer choice C have a nice day.

Use 105° = 135° - 30° to find the exact value of 105°.

Key Concepts

Angle Addition and Subtraction

Reference Angles

Exact Values of Trigonometric Functions

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