Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To find the exact value of [tex]\(\sin(345^\circ)\)[/tex], let's proceed step-by-step:
1. Recognize the reference angle:
[tex]\(345^\circ\)[/tex] is in the fourth quadrant. The reference angle is calculated by subtracting [tex]\(345^\circ\)[/tex] from [tex]\(360^\circ\)[/tex]:
[tex]\[ 360^\circ - 345^\circ = 15^\circ \][/tex]
So, the reference angle is [tex]\(15^\circ\)[/tex].
2. Use the sine value in the fourth quadrant:
In the fourth quadrant, the sine function is negative. Therefore:
[tex]\[ \sin(345^\circ) = -\sin(15^\circ) \][/tex]
3. Find the exact value of [tex]\(\sin(15^\circ)\)[/tex]:
We use the angle subtraction formula for sine:
[tex]\[ \sin(15^\circ) = \sin(45^\circ - 30^\circ) \][/tex]
Applying the sine difference formula [tex]\(\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)\)[/tex]:
[tex]\[ \sin(15^\circ) = \sin(45^\circ)\cos(30^\circ) - \cos(45^\circ)\sin(30^\circ) \][/tex]
Using the known values of trigonometric functions:
[tex]\[ \sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(30^\circ) = \frac{1}{2} \][/tex]
Substituting these into the formula:
[tex]\[ \sin(15^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \][/tex]
Simplify the expression:
[tex]\[ \sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \][/tex]
4. Determine the final expression for [tex]\(\sin(345^\circ)\)[/tex]:
Since [tex]\(\sin(345^\circ) = -\sin(15^\circ)\)[/tex]:
[tex]\[ \sin(345^\circ) = -\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right) = -\frac{\sqrt{6} - \sqrt{2}}{4} \][/tex]
We can also rewrite this as:
[tex]\[ \sin(345^\circ) = \frac{-\sqrt{6} + \sqrt{2}}{4} \][/tex]
So, the exact value of [tex]\(\sin(345^\circ)\)[/tex] is:
[tex]\[ \boxed{\frac{-\sqrt{6} + \sqrt{2}}{4}} \][/tex]
1. Recognize the reference angle:
[tex]\(345^\circ\)[/tex] is in the fourth quadrant. The reference angle is calculated by subtracting [tex]\(345^\circ\)[/tex] from [tex]\(360^\circ\)[/tex]:
[tex]\[ 360^\circ - 345^\circ = 15^\circ \][/tex]
So, the reference angle is [tex]\(15^\circ\)[/tex].
2. Use the sine value in the fourth quadrant:
In the fourth quadrant, the sine function is negative. Therefore:
[tex]\[ \sin(345^\circ) = -\sin(15^\circ) \][/tex]
3. Find the exact value of [tex]\(\sin(15^\circ)\)[/tex]:
We use the angle subtraction formula for sine:
[tex]\[ \sin(15^\circ) = \sin(45^\circ - 30^\circ) \][/tex]
Applying the sine difference formula [tex]\(\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)\)[/tex]:
[tex]\[ \sin(15^\circ) = \sin(45^\circ)\cos(30^\circ) - \cos(45^\circ)\sin(30^\circ) \][/tex]
Using the known values of trigonometric functions:
[tex]\[ \sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(30^\circ) = \frac{1}{2} \][/tex]
Substituting these into the formula:
[tex]\[ \sin(15^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \][/tex]
Simplify the expression:
[tex]\[ \sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \][/tex]
4. Determine the final expression for [tex]\(\sin(345^\circ)\)[/tex]:
Since [tex]\(\sin(345^\circ) = -\sin(15^\circ)\)[/tex]:
[tex]\[ \sin(345^\circ) = -\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right) = -\frac{\sqrt{6} - \sqrt{2}}{4} \][/tex]
We can also rewrite this as:
[tex]\[ \sin(345^\circ) = \frac{-\sqrt{6} + \sqrt{2}}{4} \][/tex]
So, the exact value of [tex]\(\sin(345^\circ)\)[/tex] is:
[tex]\[ \boxed{\frac{-\sqrt{6} + \sqrt{2}}{4}} \][/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
