91Ó°ÊÓ

The Pythagorean Identity is one of the foundational identities in trigonometry and it expresses the relationship between the sine and cosine of an angle. The identity is given as: \[ \sin^2 x + \cos^2 x = 1. \]This formula helps us to find one trigonometric function if we know the other. For example, if you know the cosine of an angle, you can determine the sine using this identity. Just rearrange the formula: \[ \sin^2 x = 1 - \cos^2 x. \]Then, you can take the square root to find the sine. However, you should always consider the quadrant in which the angle lies, as this affects the sign of the values. In our exercise, since Angle \( A \) is in the third quadrant, its sine was negative as expected.

The Angle Addition Formula for sine is crucial for finding the sine of the sum of two angles. It is represented as:\[ \sin(A + B) = \sin A \cos B + \cos A \sin B. \]This formula enables the addition of two separate angle measures to get the resultant sine value. Similarly, there is an Angle Subtraction Formula:\[ \sin(A - B) = \sin A \cos B - \cos A \sin B. \]These formulas are not limited to sine alone. They can also be applied to cosine and tangent, but it all boils down to knowing the initial sine and cosine values. In the example problem, using these formulas allows us to compute \( \sin(A + B) \) and \( \sin(A - B) \) precisely without needing a calculator.

Determining the quadrant is key to understanding the signs of trigonometric functions. Quadrants in the Cartesian plane follow a specific order: - First Quadrant: \( 0 \) to \( \frac{\pi}{2} \), both sine and cosine are positive. - Second Quadrant: \( \frac{\pi}{2} \) to \( \pi \), sine is positive, but cosine is negative. - Third Quadrant: \( \pi \) to \( \frac{3\pi}{2} \), both sine and cosine are negative. - Fourth Quadrant: \( \frac{3\pi}{2} \) to \( 2\pi \), sine is negative, but cosine is positive.In our exercise, the given ranges for \( A \) and \( B \) specify that they lie in the third quadrant. For the resulting angle sums and differences like \( A + B \) and \( A - B \), we evaluate the trigonometric results to determine their respective quadrants. This lets us understand why their sine, cosine, or tangent values have specific signs.

The tangent of an angle can be described in terms of sine and cosine:\[ \tan x = \frac{\sin x}{\cos x}. \]This formula shows how tangent relates to the ratio of sine to cosine. When faced with angle addition or subtraction, the formulas are:\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]and\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}. \]These tangent identities allow us to calculate the tangent of compound angles. It's essential to recognize how signs of the tangent might also hint on which quadrant the angle lies. In our example, these formulas help determine \( \tan(A + B) \) and \( \tan(A - B) \), which play a role in understanding where the angles reside in the coordinate plane according to their resulting signs.