Find exact value of sin (pi/4) and explain how you got the answer
The Correct Answer and Explanation is:
The exact value of sin(π/4) is:sin(π4)=22\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}sin(4π)=22
Explanation:
To find the exact value of sin(π4)\sin(\frac{\pi}{4})sin(4π), first understand what this expression means. The angle π4\frac{\pi}{4}4π radians is equivalent to 45 degrees because:π4×180∘π=45∘\frac{\pi}{4} \times \frac{180^\circ}{\pi} = 45^\circ4π×π180∘=45∘
So, the problem is asking for the sine of 45 degrees.
The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. This can be remembered using the mnemonic SOH from SOHCAHTOA:
Sine = Opposite / Hypotenuse
To find the exact sine value of 45°, imagine or construct a 45°–45°–90° triangle. This is a special type of right triangle where the two non-right angles are equal. Because the angles are equal, the legs of the triangle are also equal in length.
Assume each leg of this triangle has a length of 1 unit. By the Pythagorean Theorem:Hypotenuse=12+12=2\text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2}Hypotenuse=12+12=2
Now, using the sine definition:sin(45∘)=OppositeHypotenuse=12\sin(45^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}}sin(45∘)=HypotenuseOpposite=21
This expression can be rationalized by multiplying the numerator and denominator by 2\sqrt{2}2:12×22=22\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}21×22=22
Therefore,sin(π4)=22\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}sin(4π)=22
This is an exact value, commonly memorized as one of the standard trigonometric values on the unit circle.
