Determine whether each statement is true or false. If false, tell...

Question

Determine whether each statement is true or false. If false, tell why. See Example 4.

cos(30° + 60°) = cos 30° + cos 60°

Accepted Answer

Hey, everyone here, we are asked to find out if the given statement is true or false. And if the statement is false, we need to state the reason why here we are given sign of 15 degrees plus 75 degrees is equivalent to sign of 15 degrees plus sign of 75 degrees. Here we have four answer choice options, answer choice, A true and answer choices B through D which state that our given statement is false because the left hand side is equal to one. Whereas the right hand side is equal to either the square root of six divided by four, the square root of six divided by two or the square root of three divided by two. So in order to determine if our statement is true or false, we first need to find the value of the left hand side and the right hand side of our given equation individually. So here we can begin with the left hand side where first we just need to recall that we are given sign of 15 degrees plus 75 degrees. And now, in order to find the value of this expression, we first just need to sum the degree values. And so we have sign of 90 degrees and now we just recall that sign of 90 degrees is equivalent to one. So now we can move on to finding the value of the right hand side. So we first just need to recall that for the right hand side, we are given sign of 15 degrees plus sign of 75 degrees. And now to find the value of the right hand sign, we first need to find the value of sign of 15 degrees and sign of 75 degrees individually. So first, we can start by finding sign of 15 degrees. And so the first step in finding its value is to rewrite 15 degrees as the difference between two degree values for which we already know the sign value of. So here we can rewrite sign of 15 degrees as sign of 45 degrees minus 30 degrees. And now to find the value of our rewritten expression here, we need to recall the property where we have S of A minus B is equivalent to sine of A multiplied by cosine of B minus cosine of a multiplied by sine of B. And now utilizing this property, we can rewrite our current expression by first noticing that A is going to be 45 degrees and B will be 30 degrees. So now rewriting our expression, we have sign of 45 degrees multiplied by cosine of 30 degrees minus cosine of 45 degrees multiplied by sine of 30 degrees. And now we just need to recall each of these sine and cosine values. So we can begin with sine of 45 degrees, which we know is equivalent to cosine of 45 degrees. And both of these values are the same and they are both one divided by the square root of two. Now finding cosine of 38 degrees, we can recall this value is the square root of three divided by two. And lastly with sine of 30 degrees, this expression is equivalent to just one half. And now we just want to rewrite our current expression utilizing these identified values. And so we have one divided by the square root of two multiplied by the square root of three divided by two minus one divided by the square root of two multiplied by one half. And now just solving or simplifying this expression, we first just want to multiply the necessary values together. And so we have the square root of three divided by two, multiplied by the square root of two. And then we have minus one divided by two multiplied by the square root of two. And now we can notice that for both of our fractions here, we have the same denominator. So we can go ahead and combine the numerators. And so we have the square root of three minus one all divided by two multiplied by the square root of two. And now to simplify further, we just want to rationalize the denominator. So that means we just need to multiply the numerator and the denominator by the square root of two. And now just simplifying we are left with the square root of six minus the square root of two all divided by four. And with this, we have found our value for sign of 15 degrees. So we can now move on to finding sign of 75 degrees. And so repeating a similar process as we did with sign of 15 degrees, we now rewrite sign of 75 degrees as sign of 45 degrees plus 30 degrees. And now to find the value of this expression, we can utilize the same formula we used earlier. However, now we can notice that we are adding instead of subtracting. So now rewriting our current expression, we have sine of 45 degrees multiplied by cosine of 30 degrees minus or sorry plus we are adding in this expression here. So we have plus cosine of 45 degrees multiplied by sine of 30 degrees. And now, just as we did earlier, we need to replace each of these sine and cosine terms with their equivalent value. And so we have one divided by the square root of two multiplied by the square root of three, divided by two plus one, divided by the square root of two multiplied by one half. And now simplifying just as we did earlier, we now have the square root of six plus the square root of two all divided by four. So now we have found values for a sign of 15 degrees and sign of 75 degrees. So we can go ahead and go back to our expression for the right hand side of the equation and substitute in these values. And so we have the square root of six minus the square root of two, all divided by four plus the square root of six plus the square root of two, all divided by four. And now we can notice that both fractions have the same denominator. So we can go ahead and combine the numerators. And so simplifying we have two multiplied by the square root of six all divided by four. And now simplifying further, we see we have a two in the numerator as well as four in the denominator, which has a factor of two. So we can simplify this expression by rewriting it as the square root of six divided by two. So now we have found that the right hand side of our expression is equivalent to the square root of six divided by two. And recalling from earlier, we know that the left hand side is simply equivalent to one. So here we can clearly see that the left hand side of the equation is not equivalent to the right hand side of the equation. Therefore, we know that our statement is false. And so we are left here with answer choice c where again our statement is false because sign of 15 degrees plus 75 degrees is equivalent to one. Whereas sign of 15 degrees plus sign of 75 degrees is equivalent to the square root of six divided by two. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Determine whether each statement is true or false. If false, tell why. See Example 4. cos(30° + 60°) = cos 30° + cos 60°

Key Concepts

Cosine Addition Formula

Trigonometric Identities

Evaluating Trigonometric Functions

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