Problem 13 - Contemporary Precalculus

Question

Use the half-angle identities to evaluate the given expression exactly. $$\sin \frac{3 \pi}{8}$$

Step-by-Step Solution

Answer

Answer: The exact value of $$\sin \frac{3 \pi}{8}$$ using half-angle identities is $$\sqrt{ \frac{2 + \sqrt{2}}{4} }$$.

Step 1: Write the angle as half of another angle

First, we express the given angle as half of another angle: $$\frac{3 \pi}{8} = \frac{1}{2} \cdot \frac{3 \pi}{4}$$.

Step 2: Use the half-angle identity for sine

Now we can apply the half-angle identity for sine for the angle $$\frac{3 \pi}{4}$$: $$\sin \left( \frac{\Theta}{2} \right) = \pm \sqrt{ \frac{1 - \cos \Theta}{2} }$$

Step 3: Determine the sine of the angle in the identity

Since $$\frac{3 \pi}{8}$$ is located in the second quadrant where sine is positive, we have: $$\sin \frac{3 \pi}{8} = \sqrt{ \frac{1 - \cos \frac{3 \pi}{4}}{2} }$$

Step 4: Find the cosine of the related angle

Next, we find the cosine of the related angle $$\frac{3 \pi}{4}$$, which is in the second quadrant and has a reference angle of $$\frac{\pi}{4}$$: $$\cos \frac{3 \pi}{4} = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$

Step 5: Substitute the cosine value and simplify

Now, we substitute the value of the cosine into the half-angle identity formula we derived in step 3: $$\sin \frac{3 \pi}{8} = \sqrt{ \frac{1 - \left(-\frac{\sqrt2}{2}\right)}{2} }$$

Step 6: Simplify the expression

Finally, we simplify the expression to obtain the exact value of the sine: $$\sin \frac{3 \pi}{8} = \sqrt{ \frac{1 + \frac{\sqrt{2}}{2}}{2} } = \sqrt{ \frac{2 + \sqrt{2}}{4} }$$ Thus, the exact value of $$\sin \frac{3 \pi}{8}$$ using the half-angle identities is $$\sqrt{ \frac{2 + \sqrt{2}}{4} }$$.

Key Concepts

Half-Angle Formulas

The Half-Angle Formulas are a set of trigonometric identities that allow us to find the trigonometric function values of angles that are half of known angles. These formulas are especially useful when the exact value of a trigonometric function for the given angle is not readily found from the unit circle.
To use the Half-Angle Formulas effectively, you need to recognize the angle as half of a known angle. For example, in the problem where we evaluated \(\sin \frac{3\pi}{8}\), we first expressed it as \(\frac{1}{2} \cdot \frac{3\pi}{4}\).
The formula for the sine function is:

• \(\sin \left( \frac{\Theta}{2} \right) = \pm \sqrt{\frac{1 - \cos \Theta}{2}}\)

The plus or minus sign depends on the quadrant in which the half-angle lies. Understanding and applying these formulas can be incredibly helpful when solving for exact trigonometric values.

Sine Function

The Sine Function is one of the core trigonometric functions. It is defined as the ratio of the length of the opposite side of a right triangle to the hypotenuse.
In the context of the unit circle, the sine of an angle gives the y-coordinate of the point where the terminal side of the angle intersects the circle. This means, for any angle, \(\Theta\), its sine value reflects its vertical positioning on the circle surface.

• The Sine function is periodic, with a period of \(2\pi\).
• It is positive in the first and second quadrants and negative in the third and fourth quadrants.
• The function has a range between -1 and 1, which means \(-1 \leq \sin \Theta \leq 1\).

In the problem, we calculated \(\sin \frac{3\pi}{8}\), which is in the second quadrant of the unit circle, where sine values are positive. This characteristic helps in determining the right sign while using trigonometric identities like the Half-Angle Formulas.

Quadrants of the Unit Circle

The Unit Circle is a critical concept in trigonometry, especially when learning about functions and transformations. It is a circle with a radius of 1 centered at the origin of the coordinate plane. As an angle increases, its terminal side will rotate counter-clockwise, positioning itself in one of the four quadrants.
Each quadrant has specific characteristics:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, but cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, but cosine is positive.

Knowing the quadrant an angle lies in helps determine the sign of its sine, cosine, or tangent values. As seen in our exercise, the angle \(\frac{3\pi}{8}\) lies in the second quadrant, directing us to use the positive square root in the Half-Angle Formula. Understanding these quadrants ensures you correctly apply trigonometric identities and maximize accuracy while solving for angles.