Problem 115 - 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. $$ 150^{\circ} $$

Step-by-Step Solution

Answer

Reference angle: \(30^\circ\); Quadrant: 2; Sine: \(\frac{1}{2}\), Cosine: \(-\frac{\sqrt{3}}{2}\).

Step 1: Determine the Reference Angle

A reference angle is the acute angle that the terminal side makes with the x-axis. For angle \(150^\circ\), subtract \(150^\circ\) from \(180^\circ\) since it lies in the second quadrant: \[Reference\ Angle = 180^\circ - 150^\circ = 30^\circ.\]

Step 2: Identify the Quadrant

The angle \(150^\circ\) lies between \(90^\circ\) and \(180^\circ\). Therefore, it is in the second quadrant, where sine is positive and cosine is negative.

Step 3: Find Sine and Cosine Values

For a \(30^\circ\) reference angle in the second quadrant, use the unit circle values where \[\sin(150^\circ) = \sin(30^\circ) = \frac{1}{2} \]and\[\cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}.\]

Key Concepts

Reference Angle

A reference angle is a simple yet powerful concept in trigonometry. It is always the smallest angle that an angle makes with the x-axis. To find the reference angle, first determine which quadrant your given angle is in. For instance, with the angle \(150^\circ\), since it is in the second quadrant, you subtract it from \(180^\circ\):

• Calculate: \(180^\circ - 150^\circ = 30^\circ\).
• The reference angle is then \(30^\circ\), which is always acute (less than \(90^\circ\)).

Knowing the reference angle is crucial because it enables you to find other trigonometric values more easily.

Quadrant

Understanding quadrants helps you determine the signs of trigonometric functions like sine and cosine. The coordinate plane is divided into four quadrants:

• The first quadrant: \(0^\circ\leq\theta <90^\circ\).
• The second quadrant: \(90^\circ\leq\theta <180^\circ\).
• The third quadrant: \(180^\circ\leq\theta <270^\circ\).
• The fourth quadrant: \(270^\circ\leq\theta <360^\circ\).

For \(150^\circ\), the angle is in the second quadrant. In this quadrant, the sine value is positive while the cosine value is negative. This is important for ensuring you assign the correct signs when finding trigonometric functions using the reference angle.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions representing the vertical and horizontal components of an angle in a right triangle. When you know the reference angle and the quadrant, you can find their values:

• For \(150^\circ\), with the reference angle \(30^\circ\):
  • \(\sin(150^\circ) = \sin(30^\circ) = \frac{1}{2}\)
• The cosine is affected by the sign in the second quadrant:
  • \(\cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}\)

These are derived using the unit circle, which provides standard values for common angles.

Unit Circle

The unit circle is a crucial tool in trigonometry, used to find sine and cosine values of angles. It is a circle with a radius of one, centered at the origin of a coordinate plane. The circle's circumference passes through the coordinates \((1,0)\), indicating the starting point for measuring angles.When dealing with angles such as \(150^\circ\), the unit circle allows you to quickly find sine and cosine because each point on the circle represents \((\cos \theta, \sin \theta)\). For example, on the unit circle:

• The terminal side of \(150^\circ\) is in the second quadrant, matching the reference angle \(30^\circ\).
• The corresponding values at \(150^\circ\) are found directly or can be calculated using known values:
  • \(\sin(150^\circ) = \frac{1}{2}\)
  • \(\cos(150^\circ) = -\frac{\sqrt{3}}{2}\)

The unit circle simplifies finding trig functions, especially for common angles.