91Ó°ÊÓ

The secant function, also known as sec, is a key trigonometric function. It is closely tied to the cosine function. In fact, the secant function is the reciprocal of the cosine function. This means that for any angle \( \theta \), \( \sec \theta = \frac{1}{\cos \theta} \). This relationship helps us understand the behavior and values of the secant function across different angles.

When dealing with secant, it's essential to think about its dependence on cosine. If cosine is positive, secant will also be positive because the reciprocal of a positive number is positive. Conversely, if cosine is negative, secant will also be negative. This direct relationship makes it easier to predict the sign of secant based on where the angle lies on the unit circle.

The tangent function, or tan, is another fundamental trigonometric function. It is defined as the ratio of the sine function to the cosine function. Mathematically, it is expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This function tells us how sin and cos interact at a given angle and can change sign depending on which quadrant the angle is in.

Because both the sine and cosine change signs as the angle moves through different quadrants, the sign of tangent can be determined by observing these changes. In the first and third quadrants, where both sin and cos are of the same sign, the tangent function is positive. In the second and fourth quadrants, where sin and cos have opposite signs, the tangent function is negative. This unique behavior makes it important to always consider the quadrant when determining the sign of the tangent function.

Angles on the unit circle are divided into four quadrants. Each quadrant affects the signs of trigonometric functions in a particular way. Quadrants are defined as follows:

• First Quadrant (0° to 90°): All trigonometric functions are positive.
• Second Quadrant (90° to 180°): Sine is positive while cosine and tangent are negative.
• Third Quadrant (180° to 270°): Tangent is positive while sine and cosine are negative.
• Fourth Quadrant (270° to 360°): Cosine is positive while sine and tangent are negative.

Understanding which quadrant an angle lies in helps us determine the sign of the trigonometric functions. This is because the functions' values depend heavily on their associated signs in these quadrants. Recognizing this pattern is crucial when solving trigonometric exercises.