Domanda del libro di testo
In Exercises 1–60, verify each identity.
tan (-x) cos x = -sin x
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.5.31
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.5.31Capitolo 3, Problema 3.5.31
Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan(x/2) = √3
Guida verificata passo dopo passo1
Rewrite the equation clearly: \( \tan\left(\frac{x}{2}\right) = \sqrt{3} \).
Recall that \( \tan(\theta) = \sqrt{3} \) at specific standard angles. Identify the general solutions for \( \theta \) where \( \tan(\theta) = \sqrt{3} \).
Since \( \theta = \frac{x}{2} \), express the solutions for \( x \) by multiplying the general solutions for \( \theta \) by 2.
Use the periodicity of the tangent function, which has period \( \pi \), to write the general solution for \( \theta \) as \( \theta = \frac{\pi}{3} + k\pi \), where \( k \) is any integer.
Apply the interval restriction \( x \in [0, 2\pi) \) to find all valid values of \( x \) by substituting \( k \) values and checking which \( x \) values fall within the interval.
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Solving Trigonometric Equations
Solving trigonometric equations involves finding all angle values within a given interval that satisfy the equation. This often requires isolating the trigonometric function and using known values or identities to determine possible solutions.
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Percorso guidato
4:34
How to Solve Linear Trigonometric Equations
Tangent Function and Its Values
The tangent function, tan(θ), is periodic with period π and relates the ratio of sine to cosine. Knowing key tangent values, such as tan(π/3) = √3, helps identify solutions to equations involving tangent.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Multiple-Angle Equations and Interval Restrictions
When the angle inside the trigonometric function is a multiple of the variable (e.g., tan(x/2)), solutions must be found by considering the function's period and the specified interval, ensuring all valid solutions within [0, 2π) are included.
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4:34How to Solve Linear Trigonometric Equations
Pratica correlata
Domanda del libro di testo
Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan 3x = (√3)/3
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Domanda del libro di testo
Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅).
cot(3θ/2) = ﹣√3
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Domanda del libro di testo
Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). sec(3θ/2) = - 2
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Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 2 cos² x + sin x - 1 = 0
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Domanda del libro di testo
In Exercises 47–54, use the figures to find the exact value of each trigonometric function. cos(α/2)
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