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Given that cos π is equal to negative root five over three, where π is between zero and π, and cos π is equal to root two over three, where π is between zero and π, find the exact value of cos π plus π.
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Our goal is to find the value of cos π plus π.
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You might be tempted to use the inverse cosine function or arc cosine function to find the values of π and π and then add these two values and find the cosine of the sum.
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But weβre looking for an exact value, which your calculator may not give you if you use this method.
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A better method is to use the multiple angle formula, which gives the cosine of a sum of two angles in terms of the cosine and sine of those two angles.
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We want to find cos π plus π, so we can set π΄ equal to π and π΅ equal to π.
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And itβs looking hopeful because weβre given the values of cos π and cos π in the question.
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However, weβre not explicitly told the values of either sin π or sin π.
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Weβre going to have to work them out.
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Letβs start by finding sin π.
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Again, we might be tempted to use our calculators to find the value of π.
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That would be inverse cosine of root two over three.
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And then having π, we could then use our calculators again to find the sin of π.
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However, weβre looking for an exact value at the end, and thereβs no guarantee that our calculator will give us the exact value.
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Instead, weβll use the fact that cos of π squared plus sin of π squared is equal to one.
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This is true for any angle, and so itβs certainly true for π.
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We know that cos π is root two over three, and so cos π squared is root two over three squared.
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And root two over three squared is two over nine.
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Rearranging, we find that sin of π squared is seven over nine, and so sin π is equal to plus or minus root seven over nine.
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We have two possible values for sin π, and weβre going to have to decide between them.
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Otherwise, weβre going to end up with multiple possible values of cos π plus π.
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We do this by using our extra bit of information that π is between zero and π.
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And either by thinking about the unit circle or the graph of π¦ equals sin π₯, we can see that the sine function is positive on this interval from zero to π radians.
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Sin π is therefore positive, and so it must be root seven over nine and not negative root seven over nine.
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And we know that we can rewrite this as root seven over three.
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So now that we found the value of sin π, we can move on to finding the value of sin π.
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And we find this value in exactly the same way as we found the value of sin π.
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We write down the relationship between sin π and cos π, substitute in the value of cos π that we have from the question, and do some algebra to find that sin π is either two over three or negative two over three.
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And just as before, because π is between zero and π, sin π will be positive or at least nonnegative.
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And so sin π must be two over three and not negative two over three.
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Having found the value of sin π, we now have the values of cos π and cos π from the question and sin π and sin π which weβve worked out.
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And so weβre ready to substitute into our formula.
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Cos π is root two over three, cos π is negative root five over three, sin π is root seven over three, and sin π is two over three.
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The rest is just arithmetic.
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We find the two products using the fact that root two times root five is root 10, to get negative root 10 over nine minus two root seven over nine.
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And swapping the order of these two terms and then combining them into one fraction using the fact that they have a common denominator, we get negative two root seven plus root 10 over nine.