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.

Key Concepts

Quadrant Analysis

Quadrant analysis involves understanding which trigonometric functions are positive or negative based on the angle’s quadrant. For example, the given conditions that ? is in the third quadrant (where both sine and cosine are negative) and ? is in the second quadrant (where sine is positive and cosine is negative) are essential for determining the correct signs of tan ? and tan ?.

Tangent Difference Identity

The tangent difference identity, tan(? - ?) = (tan ? - tan ?) / (1 + tan ? tan ?), is a trigonometric formula that expresses the tangent of the difference of two angles in terms of the tangents of the individual angles. This identity is useful for breaking down complex expressions into simpler components when the individual tangent values are easier to find.

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is a fundamental relation in trigonometry that allows the calculation of an unknown trigonometric function when one of sin ? or cos ? is known. It is particularly useful when combined with quadrant information to assign the proper sign to the computed value.

Exact Trigonometric Values

Determining exact trigonometric values involves using known ratios and identities, rather than approximating with decimals. This method allows for precise outcomes expressed in fractional or radical form, which is critical in solving trigonometric expressions exactly.

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