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 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 writing down the given equation to verify: \(\frac{\sin(s + t)}{\cos s} \cot t = \tan s + \tan t\).
Recall the angle sum identity for sine: \(\sin(s + t) = \sin s \cos t + \cos s \sin t\). Substitute this into the left-hand side (LHS) of the equation.
Rewrite \(\cot t\) as \(\frac{\cos t}{\sin t}\) and substitute it into the LHS, so the expression becomes \(\frac{\sin s \cos t + \cos s \sin t}{\cos s} \times \frac{\cos t}{\sin t}\).
Simplify the LHS by distributing and canceling terms where possible. Break the fraction into two parts: \(\frac{\sin s \cos t}{\cos s} \times \frac{\cos t}{\sin t} + \frac{\cos s \sin t}{\cos s} \times \frac{\cos t}{\sin t}\).
Express the resulting terms in terms of tangent functions, using \(\tan x = \frac{\sin x}{\cos x}\), and simplify to show that the LHS equals the right-hand side (RHS), \(\tan s + \tan t\), thus 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 involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides simplify to the same expression, often by using fundamental identities like Pythagorean, reciprocal, or quotient identities.
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Fundamental Trigonometric Identities
Quotient and Reciprocal Identities
Quotient identities express tangent and cotangent in terms of sine and cosine: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. Reciprocal identities relate sine, cosine, and their reciprocals cosecant and secant. These are essential for rewriting expressions to simplify or prove identities.
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Sum of Angles Formulas
Sum of angles formulas provide expressions for trigonometric functions of sums, such as sin(s + t) = sin s cos t + cos s sin t. These formulas are crucial for expanding or simplifying expressions involving sums of angles to verify identities.
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