Find the exact value of sin(5pi/8) by using the formulas (sum and difference, half angle, double angle
The Correct Answer and Explanation is:
To find the exact value of sin(5π8)\sin\left(\frac{5\pi}{8}\right)sin(85π), we will use a combination of sum and difference identities and half-angle formulas.
Step 1: Express 5π8\frac{5\pi}{8}85π as a sum of known angles
We can write 5π8\frac{5\pi}{8}85π as the sum of two known angles: 5π8=π2+π8\frac{5\pi}{8} = \frac{\pi}{2} + \frac{\pi}{8}85π=2π+8π
Step 2: Apply the sum formula for sine
We use the sum formula for sine: sin(A+B)=sinAcosB+cosAsinB\sin(A + B) = \sin A \cos B + \cos A \sin Bsin(A+B)=sinAcosB+cosAsinB
In this case, A=π2A = \frac{\pi}{2}A=2π and B=π8B = \frac{\pi}{8}B=8π. Substituting these values into the formula: sin(π2+π8)=sin(π2)cos(π8)+cos(π2)sin(π8)\sin\left(\frac{\pi}{2} + \frac{\pi}{8}\right) = \sin\left(\frac{\pi}{2}\right)\cos\left(\frac{\pi}{8}\right) + \cos\left(\frac{\pi}{2}\right)\sin\left(\frac{\pi}{8}\right)sin(2π+8π)=sin(2π)cos(8π)+cos(2π)sin(8π)
Step 3: Simplify using known values for sin(π2)\sin\left(\frac{\pi}{2}\right)sin(2π) and cos(π2)\cos\left(\frac{\pi}{2}\right)cos(2π)
We know that: sin(π2)=1andcos(π2)=0\sin\left(\frac{\pi}{2}\right) = 1 \quad \text{and} \quad \cos\left(\frac{\pi}{2}\right) = 0sin(2π)=1andcos(2π)=0
So, the expression simplifies to: sin(π2+π8)=1⋅cos(π8)+0⋅sin(π8)\sin\left(\frac{\pi}{2} + \frac{\pi}{8}\right) = 1 \cdot \cos\left(\frac{\pi}{8}\right) + 0 \cdot \sin\left(\frac{\pi}{8}\right)sin(2π+8π)=1⋅cos(8π)+0⋅sin(8π)
Thus, we have: sin(5π8)=cos(π8)\sin\left(\frac{5\pi}{8}\right) = \cos\left(\frac{\pi}{8}\right)sin(85π)=cos(8π)
Step 4: Use the half-angle formula to find cos(π8)\cos\left(\frac{\pi}{8}\right)cos(8π)
Now, we need to compute cos(π8)\cos\left(\frac{\pi}{8}\right)cos(8π). To do this, we use the half-angle formula for cosine: cos(θ2)=±1+cos(θ)2\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}cos(2θ)=±21+cos(θ)
Let θ=π4\theta = \frac{\pi}{4}θ=4π, so that π8=π4×12\frac{\pi}{8} = \frac{\pi}{4} \times \frac{1}{2}8π=4π×21. Then: cos(π8)=1+cos(π4)2\cos\left(\frac{\pi}{8}\right) = \sqrt{\frac{1 + \cos\left(\frac{\pi}{4}\right)}{2}}cos(8π)=21+cos(4π)
We know that: cos(π4)=22\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}cos(4π)=22
Substitute this into the half-angle formula: cos(π8)=1+222=22+222=2+24=2+22\cos\left(\frac{\pi}{8}\right) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \frac{\sqrt{2 + \sqrt{2}}}{2}cos(8π)=21+22=222+22=42+2=22+2
Final Answer:
Thus, the exact value of sin(5π8)\sin\left(\frac{5\pi}{8}\right)sin(85π) is: sin(5π8)=2+22\sin\left(\frac{5\pi}{8}\right) = \frac{\sqrt{2 + \sqrt{2}}}{2}sin(85π)=22+2
