Problem 119 - Precalculus

Question

For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places. $$ \frac{3 \pi}{4} $$

Step-by-Step Solution

Answer

Reference angle: \(\frac{\pi}{4}\); Quadrant: 2; \(\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}\), \(\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\).

Step 1: Identify the Reference Angle

The reference angle for any angle is the smallest positive angle formed by the terminal side of the given angle and the x-axis. For an angle \( \theta = \frac{3 \pi}{4} \), the reference angle can be found by subtracting it from \( \pi \) (since \( \frac{3 \pi}{4} \) is in the second quadrant). The reference angle is \( \pi - \frac{3 \pi}{4} = \frac{\pi}{4} \).

Step 2: Determine the Quadrant of the Angle

To determine the quadrant in which the angle \( \frac{3 \pi}{4} \) lies, we observe that the angle is greater than \( \frac{\pi}{2} \) and less than \( \pi \). Therefore, it is in the second quadrant.

Step 3: Find Sine and Cosine of the Reference Angle

Since \( \frac{\pi}{4} \) is one of the reference angles on the unit circle, we can directly find its sine and cosine values. For \( \frac{\pi}{4} \), \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) and \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

Step 4: Adjust Sine and Cosine for the Second Quadrant

In the second quadrant, sine is positive and cosine is negative. Therefore, for the angle \( \frac{3 \pi}{4} \), the sine is positively \( \sin\left(\frac{3 \pi}{4}\right) = \frac{\sqrt{2}}{2} \) and the cosine is negatively \( \cos\left(\frac{3 \pi}{4}\right) = -\frac{\sqrt{2}}{2} \).

Key Concepts

Reference Angle

In trigonometry, the reference angle is a crucial concept that helps simplify calculations. It is the smallest positive angle made between the terminal side of a given angle and the x-axis. This angle is always positive and always between 0 and 90 degrees or, in radians, between 0 and \( \frac{\pi}{2} \).

• The reference angle is the 'base' angle we use to find core trigonometric values.
• For angles greater than \( \pi \) (or 180°), or negative angles, the reference angle is always measured to the x-axis.

To calculate reference angles:

• If the angle is in the first quadrant, it is the reference angle itself.
• In the second quadrant, the reference angle is \( \pi - \theta \).
• For the third quadrant, it's \( \theta - \pi \).
• In the fourth, \( 2\pi - \theta \).

Knowing how to find a reference angle helps us use the unit circle to derive the sine and cosine easily.

Quadrants

The coordinate plane is divided into four quadrants, which help us determine the signs of the sine and cosine of an angle. The quadrants are numbered counterclockwise starting from the positive x-axis.

• 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: Cosine is positive, but sine is negative.

Applying this knowledge to \( \frac{3\pi}{4} \) in the second quadrant, we know sine is positive and cosine is negative. Understanding quadrants helps in quickly assessing the sign of trigonometric functions without long calculations.

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of one centered at the origin of the coordinate system. Each point on the unit circle represents an angle from the origin, making calculation of sine and cosine convenient.

• Angles on the unit circle are measured in radians.
• The x-coordinate of a point on the unit circle is the cosine of the angle.
• The y-coordinate is the sine of the angle.

For instance, the angle \( \frac{\pi}{4} \) corresponds to the point \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \) on the unit circle. This makes it simple to find sine and cosine values. The unit circle allows for direct application of reference angles, simplifying sine and cosine determination.

Sine and Cosine Values

When dealing with trigonometric functions, knowing the sine and cosine values of specific angles is fundamental. These values are particularly easy to determine for angles found on the unit circle, such as multiples of \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{6} \).

• Sine measures the vertical distance from the x-axis to the point on the unit circle.
• Cosine measures the horizontal distance from the y-axis to the same point.
• Both sine and cosine values range from -1 to 1.

Particularly for \( \frac{\pi}{4} \), both values are \( \frac{\sqrt{2}}{2} \). In trigonometry, these simple fractions represent exact values, key for solving many problems.