Problem 5 - Algebra and Trigonometry

Question

\(\sin \frac{5 \pi}{4}=\) ______

Step-by-Step Solution

Answer

The sine of \( \frac{5 \pi}{4} \) is \( -\frac{\sqrt{2}}{2} \).

Step 1: Understand the Angle

Identify the given angle \( \frac{5 \pi}{4} \) and note that it is in radians. Convert this angle to degrees to better understand its location in the coordinate plane. \( \frac{5 \pi}{4} \) radians is equivalent to \( 225^\circ \) since \( \frac{\frac{5 \pi}{4}}{\pi} \times 180^\circ = 225^\circ \).

Step 2: Determine the Quadrant

The angle \( 225^\circ \) lies in the third quadrant of the unit circle. In this quadrant, the sine function (which is the y-coordinate of the corresponding point on the unit circle) is negative.

Step 3: Use Reference Angle

Find the reference angle for \( 225^\circ \). The reference angle is the acute angle formed with the x-axis, which is \( 225^\circ - 180^\circ = 45^\circ \). Thus, the reference angle is \( 45^\circ \) or \( \frac{\pi}{4} \) radians.

Step 4: Evaluate Sine of the Reference Angle

Recall the sine of the reference angle \( 45^\circ \) or \( \frac{\pi}{4} \) is \( \frac{1}{\sqrt{2}} \) or \( \frac{\sqrt{2}}{2} \).

Step 5: Apply the Sign

Since the original angle \( 225^\circ \) is in the third quadrant, where sine is negative, the sine value is \( -\frac{\sqrt{2}}{2} \).

Key Concepts

sine function

The sine function is one of the fundamental trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. Mathematically, for an angle θ, the sine function is expressed as \(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\). In a unit circle, which has a radius of 1, the sine of an angle gives the y-coordinate of the corresponding point on the circle. This function is periodic with a period of 360 degrees (or 2π radians) and ranges from -1 to 1.

unit circle

The unit circle is a circle with a radius of exactly 1, centered at the origin of the coordinate plane. It is crucial in trigonometry because it simplifies the definitions of trigonometric functions. In the unit circle:

• The x-coordinate of a point represents the cosine of the angle.
• The y-coordinate of a point represents the sine of the angle.

For any angle θ, the coordinates (cos(θ), sin(θ)) lie on this circle. The full circle is 360 degrees or 2π radians, and it is divided into four quadrants, each representing a range of 90 degrees (or π/2 radians).

reference angle

A reference angle is the smallest angle formed by the terminal side of the given angle and the x-axis. It is always between 0 and 90 degrees (0 to π/2 radians), making it easy to work with trig functions. For example, to find the reference angle for 225 degrees, which lies in the third quadrant:

• Subtract 180 degrees from 225 degrees.
• The reference angle is 45 degrees (or π/4 radians).

This helps simplify calculations because you can use known values of trigonometric functions for acute angles.

quadrants

The coordinate plane is divided into four quadrants, which help in determining the signs of the trigonometric functions:

• First Quadrant (0 to 90 degrees): Both x and y coordinates are positive. Sine and cosine values are positive here.
• Second Quadrant (90 to 180 degrees): x is negative, y is positive. Sine is positive, but cosine is negative.
• Third Quadrant (180 to 270 degrees): Both x and y coordinates are negative. Sine and cosine values are negative here.
• Fourth Quadrant (270 to 360 degrees): x is positive, y is negative. Cosine is positive, but sine is negative.

Understanding which quadrant an angle falls into helps determine the sign of its sine and cosine values.