How to find the exact value of cos 67.5 ?

  • Thread starter jkristia
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In summary, the conversation discusses a question in a precalc class about finding the exact value for cos (67.5) and how to simplify the solution. The conversation suggests taking the half angle of 135 and suggests squaring and simplifying the formula for cos(45+45/2) and then taking the square root. The final solution is to multiply both the top and bottom by ##\sqrt{2-\sqrt{2}}##.
  • #1
jkristia
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One of the questions in the homework for online precalc class I'm taking is to find the exact value for cos (67.5).
At first I didn't realize that I should just take the half angle of 135, so instead I tried to find the value for cos (45 + 45/2), and got stuck at this point.

attachment.php?attachmentid=44118&stc=1&d=1329679233.png


If I calculate the actual value I get the correct answer, but I'm not able to see how I can simplify this further.
I know the solution is as simple as finding the half angle value of

attachment.php?attachmentid=44119&stc=1&d=1329679233.png


Any suggestions of how I can get from my first attempt solution to the simpler (and correct) solution?
 

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  • #2
How is Cos(x) related to Cos(x+90), how is Cos(x+90) related to other trig functions?
 
  • #3
Try multiplying both the top and bottom by ##\sqrt{2-\sqrt{2}}##.
 
  • #4
>>How is Cos(x) related to Cos(x+90), how
yeah, that definitely makes it simpler

attachment.php?attachmentid=44137&stc=1&d=1329699522.png


>>Try multiplying both the top and bottom by
I have tried, but I seem to end up with a square root I can't get rid of, but I will keep trying.


Thanks for your help
Jesper
 

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  • #5
Hi, jkristia,

Square your formula for cos(45+45/2), simplify, and then take the square root.ehild
 
  • #6
ehild said:
Hi, jkristia,

Square your formula for cos(45+45/2), simplify, and then take the square root.


ehild

ah - of course, that works
Thank you very much
 

1. How do I calculate the exact value of cos 67.5?

To calculate the exact value of cos 67.5, we can use the half-angle formula for cosine: cos (2x) = 1 - 2sin^2(x). Substituting 67.5 degrees for x, we get cos (2 * 67.5) = 1 - 2sin^2(67.5). Using a scientific calculator, we can solve for sin(67.5) and then plug it back into the formula to find the exact value of cos 67.5.

2. Is there a simpler way to find the exact value of cos 67.5?

Yes, there is an alternative method using the trigonometric identity: cos (90 - x) = sin(x). In this case, we can rewrite cos 67.5 as cos (90 - 22.5) and use the fact that sin 22.5 has a known exact value of √(2 - √3)/2. Plugging this value into the identity, we can easily find the exact value of cos 67.5.

3. Can I use a calculator to find the exact value of cos 67.5?

Yes, you can use a calculator to find the exact value of cos 67.5, but it depends on the type of calculator you have. Basic calculators may not have the necessary functions to solve for trigonometric identities, so using a scientific calculator would be more helpful.

4. What is the significance of finding the exact value of cos 67.5?

Finding the exact value of cos 67.5 can be useful in various mathematical and scientific applications, such as calculating the exact distance between two points using the Law of Cosines or determining the exact angle of a vector in a physics problem. It can also help verify the accuracy of other approximations or calculations.

5. Are there any other methods to find the exact value of cos 67.5?

Yes, there are other methods to find the exact value of cos 67.5, such as using the Taylor series expansion or the De Moivre's formula. However, these methods may require more advanced knowledge of mathematics and are not commonly used for this specific calculation.

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