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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
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Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 65
Capitolo 3, Problema 65

In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 5 cos² x - 3 = 0

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Start with the given equation: \(5 \cos^{2} x - 3 = 0\).
Isolate the cosine squared term by adding 3 to both sides: \(5 \cos^{2} x = 3\).
Divide both sides by 5 to solve for \(\cos^{2} x\): \(\cos^{2} x = \frac{3}{5}\).
Take the square root of both sides to solve for \(\cos x\): \(\cos x = \pm \sqrt{\frac{3}{5}}\).
Find all values of \(x\) in the interval \([0, 2\pi)\) where \(\cos x = \sqrt{\frac{3}{5}}\) and where \(\cos x = -\sqrt{\frac{3}{5}}\), using the unit circle or inverse cosine function.

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Solving Quadratic Trigonometric Equations

This involves treating trigonometric equations like algebraic quadratics by substituting expressions such as cos²x with a variable. After solving the quadratic equation, substitute back to find the trigonometric values and then solve for the angle x within the given interval.
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Solving Quadratic Equations by Completing the Square

Unit Circle and Interval Restrictions

Understanding the unit circle helps identify all angle solutions for trigonometric equations within a specified interval, here [0, 2π). Since cosine values repeat every 2π, solutions must be found within one full rotation, considering both positive and negative cosine values.
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Introduction to the Unit Circle

Exact Values and Approximate Solutions

Exact values refer to well-known trigonometric values expressed in terms of π or simple fractions, while approximate solutions use decimal values rounded to a specified precision. Knowing when and how to use each is essential for providing answers as required by the problem.
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Example 1
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In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin² x - 2 cos x - 2 = 0

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In Exercises 59–68, verify each identity. cotx2=sinx1−cosx\(\cot\)\(\frac{x}{2}\)=\(\frac{\sin x}{1-\cos x}\)

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In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .

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In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .

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