91Ó°ÊÓ

Understanding the concept of a reference angle is key in trigonometry. A reference angle is the smallest angle that the terminal side of the given angle makes with the x-axis. It allows us to simplify the determination of trigonometric function values by relating them back to well-known angles in the first quadrant, generally between 0° and 90°.
To find the reference angle, you must consider:

• For angles in the second quadrant (90° to 180°), subtract the given angle from 180°.
• For the third quadrant (180° to 270°), subtract 180° from the angle.
• In the fourth quadrant (270° to 360°), subtract the angle from 360°.

Finding the reference angle transforms a complex scenario into something more familiar, allowing you to use known trigonometric values like \(30^{\circ}\), 45°, or 60°.

The Cartesian plane is divided into four quadrants, which are essential for understanding the signs of trigonometric functions. Each quadrant covers a 90° span of the circle, and the direction moves anti-clockwise starting from the positive x-axis:

• Quadrant I: Angles here range from 0° to 90°. All trigonometric functions (sine, cosine, tangent) are positive.
• Quadrant II: From 90° to 180°. In this quadrant, sine is positive. However, cosine and tangent are negative.
• Quadrant III: Spanning angles from 180° to 270°. Here, tangent is positive, while sine and cosine are negative.
• Quadrant IV: Covers angles from 270° to 360°. Cosine is positive in this quadrant, while sine and tangent are negative.

Recognizing the quadrant helps in determining the sign of the cosine and other trigonometric values for angles outside the first quadrant.

Calculating the cosine of an angle without a calculator involves understanding the reference angle and the properties of quadrants. Once you have the reference angle, you use its known trigonometric value. For instance, \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), clarifies what cosine value to use.
Next, apply the sign based on the angle’s quadrant:

• If the reference angle is in Quadrant I, cosine is positive.
• In Quadrant II, cosine becomes negative.
• Quadrant III places cosine also at negative.
• Finally, Quadrant IV returns cosine to positive.

So, when finding \( \cos 150^{\circ} \), you identify its reference angle \(30^{\circ}\) and know cosine there \( \frac{\sqrt{3}}{2} \). Since 150° lies in Quadrant II, apply the negative sign, resulting in \( -\frac{\sqrt{3}}{2} \). This process demonstrates how understanding the underlying principles replaces the need for computation devices.