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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 24
Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. tan ( 5𝝅/3 ﹣ 𝝅/4)
Guida verificata passo dopo passo1
Identify the given expression: \(\tan\left( \frac{5\pi}{3} - \frac{\pi}{4} \right)\).
Recall the tangent difference identity: \(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\).
Set \(A = \frac{5\pi}{3}\) and \(B = \frac{\pi}{4}\), then find \(\tan A\) and \(\tan B\) separately.
Calculate \(\tan \frac{5\pi}{3}\) and \(\tan \frac{\pi}{4}\) using known values or reference angles.
Substitute these values into the tangent difference formula and simplify the resulting expression step-by-step.
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Sum and Difference Identities for Tangent
These identities express the tangent of a sum or difference of two angles in terms of the tangents of the individual angles. Specifically, tan(A - B) = (tan A - tan B) / (1 + tan A * tan B). This formula is essential for breaking down complex tangent expressions into simpler parts.
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Percorso guidato
4:47
Sum and Difference of Tangent
Exact Values of Trigonometric Functions at Special Angles
Certain angles like π/3, π/4, and π/5 have known exact trigonometric values involving square roots and rational numbers. Knowing these values allows for precise calculation without approximations, which is crucial when solving problems requiring exact answers.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Simplification of Trigonometric Expressions
After applying identities, simplifying the resulting expressions by combining like terms, rationalizing denominators, or reducing fractions is necessary. This step ensures the final answer is in its simplest exact form, which is often required in trigonometry exercises.
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6:36Simplifying Trig Expressions
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Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°
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Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. tan ( 𝝅/3 + 𝝅/4 )
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Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). cos 4x = ﹣√3 / 2
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