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Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 9 tan² x - 3 = 0
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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 3.5.53
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.5.53Capitolo 3, Problema 3.5.53
In Exercises 53–62, solve each equation on the interval [0, 2𝝅). (tan x - 1) (cos x + 1) = 0
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Recognize that the equation is a product of two factors equal to zero: \((\tan x - 1)(\cos x + 1) = 0\). According to the zero product property, set each factor equal to zero separately: \(\tan x - 1 = 0\) and \(\cos x + 1 = 0\).
Solve the first equation \(\tan x - 1 = 0\) which simplifies to \(\tan x = 1\). Recall that \(\tan x = 1\) at angles where the sine and cosine are equal in magnitude and sign, specifically in the first and third quadrants within \([0, 2\pi)\).
Find the general solutions for \(\tan x = 1\) on the interval \([0, 2\pi)\), which correspond to \(x = \frac{\pi}{4}\) and \(x = \frac{5\pi}{4}\).
Solve the second equation \(\cos x + 1 = 0\) which simplifies to \(\cos x = -1\). Recall that cosine equals \(-1\) at the angle where the terminal side points directly to the left on the unit circle.
Find the solution for \(\cos x = -1\) on the interval \([0, 2\pi)\), which is \(x = \pi\). Combine all solutions from both equations to get the complete solution set.
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Solving Trigonometric Equations
Solving trigonometric equations involves finding all angle values within a specified interval that satisfy the equation. This often requires isolating trigonometric functions and using their known values or identities to determine solutions.
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Percorso guidato
4:34
How to Solve Linear Trigonometric Equations
Zero Product Property
The zero product property states that if the product of two factors equals zero, then at least one of the factors must be zero. This allows the equation (tan x - 1)(cos x + 1) = 0 to be split into two simpler equations: tan x - 1 = 0 and cos x + 1 = 0.
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05:40Introduction to Dot Product
Trigonometric Function Values and Unit Circle
Understanding the values of tangent and cosine functions on the unit circle is essential. For example, tan x = 1 corresponds to angles where sine and cosine are equal, and cos x = -1 corresponds to the angle where the point on the unit circle is at (-1, 0). This knowledge helps identify exact solutions within [0, 2π).
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6:34Sine, Cosine, & Tangent on the Unit Circle
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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). tan 3x = (√3)/3
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Domanda del libro di testo
In Exercises 39–46, use a half-angle formula to find the exact value of each expression. tan(7𝝅/8)
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Domanda del libro di testo
Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 sin² x = sin x + 3
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Domanda del libro di testo
In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. sin² x cos² x
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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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