91Ó°ÊÓ

Half-angle identities are invaluable tools in trigonometry for simplifying expressions involving angles that are half of a given angle. For instance, if you know the trigonometric value of an angle \( \alpha \), you can figure out the value of its half angle, \( \frac{\alpha}{2} \).
The half-angle identity for cosine is particularly useful. It can be expressed as:

• \( \cos \left( \frac{\alpha}{2} \right) = \pm \sqrt{ \frac{1 + \cos \alpha}{2} } \)

This formula simplifies the calculation of the cosine of half an angle if you have the cosine of the original angle.
However, the ‘\( \pm \)’ indicates that the result can be either positive or negative depending on the quadrant where \( \frac{\alpha}{2} \) lies. In this exercise, we determined the sign based on which trigonometric quadrant the angle \( \frac{\alpha}{2} \) falls into. In our case, it was the second quadrant, where the cosine function is negative.

The Pythagorean identity is a cornerstone of trigonometry. It connects the squares of sine and cosine of the same angle, forming a vital relationship:

• \( \sin^2 \alpha + \cos^2 \alpha = 1 \)

This identity allows us to find the value of one trigonometric function when we have the other.
In the exercise, the angle \( \alpha \) was given in the third quadrant with \( \sin \alpha = -\frac{4}{5} \). We could then find \( \cos \alpha \) using:

• \( \cos^2 \alpha = 1 - \sin^2 \alpha = 1 - \left(-\frac{4}{5}\right)^2 = \frac{9}{25} \)

Since cosine is negative in the third quadrant, \( \cos \alpha = -\frac{3}{5} \).
Thus, the Pythagorean identity helps us to bridge the gap between sine and cosine, giving us a comprehensive tool for solving trigonometric equations.

Trigonometric quadrants play a crucial role in determining the sign of trigonometric functions based on the angle's position on the coordinate plane. The plane is divided into four quadrants:

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

In our exercise, \( \alpha \) belonged to the third quadrant, which dictated the signs for \( \sin \alpha \) and \( \cos \alpha \). Therefore, understanding the quadrants helps in applying the correct signs to trigonometric values especially when using identities like the half-angle identity.