Find the exact value of the indicated expression using the...

Find the exact value of the indicated expression using the given information and identities. Find the exact value of $\tan (\alpha+\beta)$ if $\sin \alpha=-\frac{3}{5}$ and $\cos \beta=-\frac{1}{4}$ and the terminal side of $\alpha$ lies in QIII and the terminal side of $\beta$ lies in QII.

Find the exact value of the indicated expression using the given information and identities.
Find the exact value of $\tan (\alpha+\beta)$ if $\sin \alpha=-\frac{3}{5}$ and $\cos \beta=-\frac{1}{4}$ and the terminal side of $\alpha$ lies in QIII and the terminal side of $\beta$ lies in QII.

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Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry connecting the sine and cosine of an angle. It states that sin²? + cos²? = 1. This identity is used to find the missing trigonometric function (cosine from sine or vice versa) when one is known, considering the sign based on the angle's quadrant.

Quadrant Sign Conventions

Quadrant sign conventions determine whether sine, cosine, and tangent values are positive or negative based on the location of the terminal side of an angle in the coordinate plane. Understanding these conventions is crucial for correctly evaluating trigonometric functions, especially when applying the Pythagorean identity and determining the correct signs during computations for angles in different quadrants.

Angle Addition Formula for Tangent

This formula allows you to find the tangent of the sum of two angles by relating it to the tangents of the individual angles. It is essential for combining angular expressions in trigonometry and is expressed as tan(a + b) = (tan a + tan b) / (1 - tan a tan b), which helps simplify and evaluate composite angle expressions when the individual tangents are known.

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