Problem 172 - Precalculus

Question

For the following exercises, use a graphing calculator to evaluate. $$ \sin \left(\frac{-5 \pi}{4}\right) \sin \left(\frac{11 \pi}{6}\right) $$

Step-by-Step Solution

Answer

The product is \(-\frac{\sqrt{2}}{4}\).

Step 1: Find the Exact Value of \( \sin \left(\frac{-5\pi}{4}\right) \)

To find \( \sin \left(\frac{-5\pi}{4}\right) \), we first convert it into a positive angle by adding \( 2\pi \) (since the sine function is periodic with period \( 2\pi \)): \[ -\frac{5\pi}{4} + 2\pi = \frac{3\pi}{4}. \] The angle \( \frac{3\pi}{4} \) corresponds to the same position in the unit circle as the angle they reduce to in the interval \( [0, 2\pi) \). The sine value at \( \frac{3\pi}{4} \) is \( \frac{\sqrt{2}}{2} \). Therefore, \( \sin \left(\frac{-5\pi}{4}\right) = \frac{\sqrt{2}}{2}. \)

Step 2: Find the Exact Value of \( \sin \left(\frac{11\pi}{6}\right) \)

The angle \( \frac{11\pi}{6} \) is equivalent to \( 2\pi - \frac{\pi}{6} \), and it lies in the fourth quadrant where sine is negative. We know \( \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \). Therefore, \( \sin \left(\frac{11\pi}{6}\right) = -\frac{1}{2}. \)

Step 3: Multiply the Sine Values

Now that we have \( \sin \left(\frac{-5\pi}{4}\right) = \frac{\sqrt{2}}{2} \) and \( \sin \left(\frac{11\pi}{6}\right) = -\frac{1}{2} \), multiply these values: \[ \frac{\sqrt{2}}{2} \times \left(-\frac{1}{2}\right) = -\frac{\sqrt{2}}{4}. \]

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It describes the ratio of the length of the opposite side to the hypotenuse in a right triangle. This function is crucial for understanding periodic phenomena, such as sound waves and light waves, and is often used in various fields like physics, engineering, and even music.

• The sine of an angle in a right triangle is given by \[\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}.\]
• The sine function is periodic, with a period of \(2\pi\), meaning \(\sin(\theta + 2\pi) = \sin(\theta)\).
• Its values range between -1 and 1, making it a bounded function.

Grasping the sine function's behavior helps in interpreting angles and corresponding sine values for practical applications.

Unit Circle

The unit circle is essential in trigonometry for defining the trigonometric functions on the coordinate plane. It is a circle with a radius of 1, centered at the origin of the coordinate system.

• Every point \((x, y)\) on the unit circle is such that \(x^2 + y^2 = 1\).
• The angle theta, typically measured in radians, is counted from the positive x-axis.
• For any angle \(\theta\), the coordinates \((x, y)\) correspond to \((\cos(\theta), \sin(\theta))\).

Understanding the unit circle allows one to visualize how the sine function varies with angles. Using the unit circle, you can easily find sine values based on coordinate points, especially those commonly used in trigonometry, like \(\pi/4, \pi/3,\) and \(\pi/6\).

Angle Conversion

Angle conversion is transforming angles from one unit of measure to another, typically from degrees to radians or vice versa, enabling easier calculations in trigonometry.

• Radians and degrees are two units for measuring angles.
• One complete circle is \(360\) degrees, which equals \(2\pi\) radians.
• To convert degrees to radians, use the formula \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\).
• Conversely, to convert radians to degrees: \(\theta_{deg} = \theta_{rad} \times \frac{180}{\pi}\).

For example, the angle \(\frac{-5\pi}{4}\) is in radians. Sometimes, angles need to be within a certain interval, often \([0, 2\pi)\), to align with the periodicity of trigonometric functions.

Quadrants of the Circle

The coordinate plane is divided into four quadrants, each playing a crucial role in determining the signs of trigonometric functions based on the angle's position in the unit circle.

• The first quadrant has both x and y coordinates positive, making all trigonometric functions positive.
• In the second quadrant, x is negative while y remains positive; thus, sine is positive while cosine and tangent are negative.
• The third quadrant has both coordinates negative, making only the tangent positive.
• The fourth quadrant features a positive x and a negative y, resulting in a positive cosine and a negative sine.

Realizing which quadrant an angle lies in helps determine the correct sign for its trigonometric functions. For instance, \(\frac{-5\pi}{4}\) initially lies beyond the circle's typical interval, but after adjustment, it corresponds to \(\frac{3\pi}{4}\), located in the second quadrant, where sine is positive.