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What is the value of
sin((pi)/(4))?
Choose 1 answer:
(A)
(1)/(2)
(B)
(sqrt2)/(2)
(C)
(sqrt3)/(2)
(D) 45

What is the value of $\sin \left(\frac{\pi}{4}\right) ?$ $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{2}$ $\newline$ (B) $\frac{\sqrt{2}}{2}$ $\newline$ (C) $\frac{\sqrt{3}}{2}$ $\newline$ (D) $45$

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Mean, median, mode, and range: find the missing number

Full solution

Q. What is the value of

\sin \left(\frac{\pi}{4}\right) ?

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{2}

\newline

(B)

\frac{\sqrt{2}}{2}

\newline

(C)

\frac{\sqrt{3}}{2}

\newline

(D)

45

Find sine value at $\frac{\pi}{4}$ radians: We need to find the value of the sine function at an angle of $\frac{\pi}{4}$ radians. $\newline$ The sine of $\frac{\pi}{4}$ is a well-known trigonometric value. $\newline$ We can recall the value from the unit circle or from trigonometric identities.
Using the unit circle: The unit circle shows that at an angle of $\frac{\pi}{4}$ radians, both the $x$ -coordinate and the $y$ -coordinate have the same value because $\frac{\pi}{4}$ radians corresponds to a $45$ -degree angle where the triangle is isosceles. $\newline$ The sine function gives the $y$ -coordinate of the unit circle at that angle.
Using the Pythagorean theorem: Since the triangle is isosceles and the hypotenuse is $1$ (because it's a unit circle), the legs of the triangle are equal in length. $\newline$ Using the Pythagorean theorem, we can find the length of the legs: $x^2 + x^2 = 1^2$ , which simplifies to $2x^2 = 1$ .
Solving for x: Solving for x, we get $x^2 = \frac{1}{2}$ , which means $x = \sqrt{\frac{1}{2}}$ . $\newline$ This simplifies to $x = \frac{\sqrt{2}}{2}$ because $\sqrt{1} = 1$ and $\sqrt{2}$ remains under the radical.
Final result: Therefore, the value of $\sin(\frac{\pi}{4})$ is the same as the length of the leg of the triangle, which is $\frac{\sqrt{2}}{2}$ . $\newline$ This matches one of the given choices.

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