91Ó°ÊÓ

In trigonometry, the coordinate plane is divided into four sections called quadrants. Each quadrant holds a unique combination of positive and negative values for the trigonometric functions. Here's how they work:
When an angle is in:

• Quadrant I: All trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are positive.
• Quadrant II: Only sine and its reciprocal, cosecant, are positive.
• Quadrant III: Only tangent and its reciprocal, cotangent, are positive.
• Quadrant IV: Only cosine and its reciprocal, secant, are positive.

Understanding the quadrant where an angle resides helps determine the signs of the trigonometric functions.
For example, \(-57^{\text{°}}\) is converted to \()303^{\text{°}}\) which places it in Quadrant IV, where cosine and secant are positive, and the other functions are negative.

A standard position angle is one that starts at the positive x-axis and rotates counterclockwise. If the angle is negative, it rotates clockwise instead. This position helps us determine in which quadrant the angle is located.

To convert a negative angle to its equivalent positive angle, you can add multiples of \(360^{\text{°}}\) until the angle is positive. For \(-57^{\text{°}}\), adding \(360^{\text{°}}\) results in: \[360^{\text{°}} + (-57^{\text{°}}) = 303^{\text{°}}\]
This means \(-57^{\text{°}}\) in standard position is the same as \(303^{\text{°}}\) and is in the Fourth Quadrant.
Knowing how to convert and identify standard position angles is essential in trigonometry as it directly relates to determining the signs of the functions.

The signs of trigonometric functions change depending on the quadrant in which the angle resides. Here are the signs of each function based on the quadrant:

• Quadrant I: All functions are positive.
• Quadrant II: Sine and cosecant are positive; cosine, secant, tangent, and cotangent are negative.
• Quadrant III: Tangent and cotangent are positive; sine, cosine, cosecant, and secant are negative.
• Quadrant IV: Cosine and secant are positive; sine, cosecant, tangent, and cotangent are negative.

For example, in the Fourth Quadrant, for \(-57^{\text{°}}\) (or \(303^{\text{°}}\)), only cosine (cos) and secant (sec) are positive. The other functions like sine (sin), tangent (tan), cosecant (csc), and cotangent (cot) will be negative.