Exact value of Trigonometric equation.

In summary, the conversation discusses the solution to the equation 1/√2 cos15 - 1/√2 sin15, with the result ultimately being determined to be 1/2 using the auxiliary angle formula. The reduction formula is also mentioned as an alternative name for the auxiliary angle formula.
  • #1
ifomoe
5
0

Homework Statement


[tex]1/\sqrt{}2 cos 15 - 1/\sqrt{}2 sin 15[/tex]


The Attempt at a Solution


I got [tex]\sqrt{}3/2[/tex].
If anyone could confirm this, I would appreciate it. Thanks.
 
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  • #2
Welcome to PF!

ifomoe said:

Homework Statement


[tex]1/\sqrt{}2 cos 15 - 1/\sqrt{}2 sin 15[/tex]


The Attempt at a Solution


I got [tex]\sqrt{}3/2[/tex].
If anyone could confirm this, I would appreciate it. Thanks.

Hi ifomoe! Welcome to PF! :smile:

(have a square-root: √ :smile:)

(I assume you mean (1/√2)cos15 - (1/√2)sin15?)

Nooo … you've used + instead of - :cry:

Which formula did you use? :smile:
 
  • #3
Thank you. :smile:
Yes. I meant it like that: (1/√2)cos15 - (1/√2)sin15.
I used the reduction formula but I just re-did it and got 1/2.
Please, don't tell me I'm wrong again. My brain can't handle much more of this. :frown:
 
  • #4
ifomoe brain good!

ifomoe said:
Thank you. :smile:
Yes. I meant it like that: (1/√2)cos15 - (1/√2)sin15.
I used the reduction formula but I just re-did it and got 1/2.

:biggrin: Woohoo! :biggrin:

(btw, what's the "reduction formula"?

I used sin(45 - 15) = sin45 cos15 - cos45 sin15. :wink:)
 
  • #5
Yay! :smile:

I think my lecturer said that another name for reduction formula is auxiliary angle formula.
If I'm not mistaken.
 
  • #6
[tex]sin(x/2)= \sqrt{\frac{1}{2}(1- cos(x))}[/tex]
so
[tex]sin(15)= \sqrt{\frac{1}{2}(1- cos(30))}= \sqrt{\frac{1}{2}\left(1- \frac{\sqrt{3}}{2}\right)}[/tex]
 
  • #7
HallsofIvy said:
[tex]sin(x/2)= \sqrt{\frac{1}{2}(1- cos(x))}[/tex]
so
[tex]sin(15)= \sqrt{\frac{1}{2}(1- cos(30))}= \sqrt{\frac{1}{2}\left(1- \frac{\sqrt{3}}{2}\right)}[/tex]

:biggrin: Or (cosθ - sinθ)/√2 = √2 sin45 sin(45 - θ) = sin(45 - θ). :biggrin:
 
Last edited:

1. What is the exact value of a trigonometric equation?

The exact value of a trigonometric equation is a specific numerical value that satisfies the equation. This value can be calculated using trigonometric identities and properties.

2. How do you find the exact value of a trigonometric equation?

To find the exact value of a trigonometric equation, you can use a calculator or a table of trigonometric values. You can also use trigonometric identities and properties to simplify the equation and determine the exact value.

3. Can you give an example of finding the exact value of a trigonometric equation?

Sure, for example, if we have the equation sin(45°), we can use a calculator or a table to find the exact value, which is 0.70710678118. We can also use the half-angle identity for sine to simplify the equation to √(2)/2, which is the exact value.

4. Why is it important to know the exact value of a trigonometric equation?

Knowing the exact value of a trigonometric equation is important in many real-world applications, such as engineering, physics, and navigation. It allows us to make accurate calculations and predictions based on the given equation.

5. Are there any tricks or shortcuts to finding the exact value of a trigonometric equation?

Yes, there are various trigonometric identities and properties that can be used to simplify equations and find the exact value. For example, the sum and difference identities, double-angle identities, and half-angle identities can all be helpful in finding the exact value of a trigonometric equation.

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