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Find all solutions of each equation. 3 sin θ + 5 = ﹣2 sin θ
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 3, Problema 22
In Exercises 1–60, verify each identity. cot² t /csc t = csc t - sin t
Guida verificata passo dopo passo1
Start with the left-hand side (LHS) of the identity: \( \frac{\cot^{2} t}{\csc t} \). Recall the definitions of cotangent and cosecant in terms of sine and cosine: \( \cot t = \frac{\cos t}{\sin t} \) and \( \csc t = \frac{1}{\sin t} \).
Rewrite \( \cot^{2} t \) as \( \left( \frac{\cos t}{\sin t} \right)^{2} = \frac{\cos^{2} t}{\sin^{2} t} \). Substitute this and \( \csc t = \frac{1}{\sin t} \) into the LHS to get \( \frac{\frac{\cos^{2} t}{\sin^{2} t}}{\frac{1}{\sin t}} \).
Simplify the complex fraction by multiplying numerator and denominator: \( \frac{\cos^{2} t}{\sin^{2} t} \times \sin t = \frac{\cos^{2} t \cdot \sin t}{\sin^{2} t} = \frac{\cos^{2} t}{\sin t} \).
Now, focus on the right-hand side (RHS): \( \csc t - \sin t = \frac{1}{\sin t} - \sin t \). To combine these terms, write \( \sin t \) as \( \frac{\sin^{2} t}{\sin t} \) to get a common denominator: \( \frac{1}{\sin t} - \frac{\sin^{2} t}{\sin t} = \frac{1 - \sin^{2} t}{\sin t} \).
Use the Pythagorean identity \( 1 - \sin^{2} t = \cos^{2} t \) to rewrite the numerator, so the RHS becomes \( \frac{\cos^{2} t}{\sin t} \), which matches the simplified LHS, thus verifying the identity.
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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 of the equation are equivalent by using known identities and algebraic manipulation.
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Percorso guidato
5:32
Fundamental Trigonometric Identities
Reciprocal and Quotient Identities
Reciprocal identities relate sine, cosine, and tangent to their reciprocal functions cosecant, secant, and cotangent. For example, csc t = 1/sin t and cot t = cos t / sin t. These identities help rewrite expressions to simplify or verify equations.
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03:40Quotients of Complex Numbers in Polar Form
Algebraic Manipulation of Trigonometric Expressions
Simplifying or verifying identities often requires factoring, combining fractions, and using common denominators. Careful algebraic steps allow transforming one side of the equation into the other, confirming the identity's validity.
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6:36Simplifying Trig Expressions
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