Textbook Question
Use the given information to find tan(x + y).
cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 5.66
Textbook Question
Verify that each equation is an identity.
sin(s + t)/cos s cot t = tan s + tan t
Verified step by step guidance1
Start by rewriting the left side of the equation: \( \frac{\sin(s + t)}{\cos s \cot t} \).
Recall the identity for \( \sin(s + t) \): \( \sin(s + t) = \sin s \cos t + \cos s \sin t \). Substitute this into the equation.
Rewrite \( \cot t \) as \( \frac{\cos t}{\sin t} \) and substitute it into the equation.
Simplify the expression: \( \frac{\sin s \cos t + \cos s \sin t}{\cos s \cdot \frac{\cos t}{\sin t}} \) becomes \( \frac{\sin s \cos t + \cos s \sin t}{\frac{\cos s \cos t}{\sin t}} \).
Simplify further by multiplying the numerator and the denominator by \( \sin t \) to get \( \sin s \tan t + \tan s \sin t \), which simplifies to \( \tan s + \tan t \), verifying the identity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variables involved, provided the expressions are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for verifying equations and simplifying trigonometric expressions.
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Sum and Difference Formulas
Sum and difference formulas express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For example, sin(s + t) = sin s cos t + cos s sin t. These formulas are essential for manipulating and simplifying expressions involving multiple angles in trigonometric equations.
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Tangent Function and Its Relationships
The tangent function is defined as the ratio of the sine and cosine functions, tan θ = sin θ / cos θ. Additionally, the tangent of the sum of two angles can be expressed as tan(s + t) = (tan s + tan t) / (1 - tan s tan t). Recognizing these relationships is vital for transforming and verifying trigonometric identities involving tangent.
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