In Exercises 63–68, find the exact value of each expression. Do n...

Question

In Exercises 63–68, find the exact value of each expression. Do not use a calculator.

cos 12° sin 78° + cos 78° sin 12°

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following expression without the help of a calculator. The expression is the cosine of 18 degrees multiplied by the sign of 72 degrees plus the cosine of degrees multiplied by the sign of 18 degrees. Our answer choices are A zero B negative one, C one D two. Recall your co function identities. One of which is that the sign of 90 degrees minus theta, which stands for any degree measure will equal the cosine of theta. Also that the cosine of 90 degrees minus theta equals the sign of theta. So how does this help us? Well, first, if you look at our expression, you notice that 18 and 72 total to this will help us be able to substitute things in. So starting at the beginning, we have the cosine of 18 multiplied by the sign of 72. I wanna make the cosine of 18 into something in terms of sign. So I can say that the cosine of 18 degrees is equivalent to the sign of 90 degrees minus 18 degrees and 90 minus 18 is 72. So the cosine of 18 degrees is equivalent to the sign of 72 degrees. So I'm gonna substitute that in for my first term. I now have the sign of 72 degrees multiplied by the sign of 72 degrees. Now, the next part or the second term I have the cosine of 72 degrees and the sign of 18 degrees. Well, this means I could take the sign of 18 degrees and rewrite it as the cosine of 90 degrees minus 18 degrees. It would be the cosine of 72 degrees. So I'm gonna go and substitute that in for the sign of 18 degrees in my second term. So my second term will now be the cosine of 72 degrees multiplied by the cosine of 72 degrees. When a value is multiplied by itself, we can square it. So we now can say we have sine squared 72 degrees plus cosine squared 72 degrees. And if you recall the trig identities sine squared, theta plus cosine squared, theta equals one. So the value of this expression is one which is answer choice C have a nice day.

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. cos 12° sin 78° + cos 78° sin 12°

Key Concepts

Trigonometric Identities

Sine and Cosine Functions

Exact Values of Trigonometric Functions

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