Domanda del libro di testo
In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. cos² 15° - sin² 15°
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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.RE.57
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.RE.57Capitolo 3, Problema 3.RE.57
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. tan x = 2 cos x tan x
Guida verificata passo dopo passo1
Start by writing down the given equation: \(\tan x = 2 \cos x \tan x\).
Recall that \(\tan x = \frac{\sin x}{\cos x}\), so substitute this into the equation to express everything in terms of sine and cosine: \(\frac{\sin x}{\cos x} = 2 \cos x \cdot \frac{\sin x}{\cos x}\).
Simplify the right side by canceling \(\cos x\) where possible, keeping in mind the domain restrictions where \(\cos x \neq 0\) to avoid division by zero.
Rearrange the equation to isolate terms and set it equal to zero, which will allow factoring or using trigonometric identities to find solutions.
Solve the resulting equation(s) for \(x\) within the interval \([0, 2\pi)\), considering all possible cases including when \(\cos x = 0\) (since division by zero was excluded earlier).
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Trigonometric Equations
Trigonometric equations involve functions like sine, cosine, and tangent. Solving these equations means finding all angle values within a specified interval that satisfy the equation. Understanding how to manipulate and simplify these equations is essential for finding exact or approximate solutions.
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Percorso guidato
4:34
How to Solve Linear Trigonometric Equations
Interval Notation and Domain Restrictions
The problem restricts solutions to the interval [0, 2π), meaning all solutions must be found between 0 and just before 2π radians. Recognizing this domain helps limit the possible solutions and ensures answers are relevant to the given range.
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Percorso guidato
06:01i & j Notation
Relationship Between Tangent and Cosine Functions
The equation involves both tangent and cosine functions, which are related through sine and cosine (tan x = sin x / cos x). Understanding how to express tangent in terms of sine and cosine allows for algebraic manipulation and simplification, facilitating the solving process.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
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Domanda del libro di testo
In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
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Domanda del libro di testo
In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value. sin 2x - sin 4x
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Domanda del libro di testo
In Exercises 35–38, find the exact value of the following under the given conditions:
e. cos(β/2)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
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Domanda del libro di testo
In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
1047
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Domanda del libro di testo
In Exercises 35–38, find the exact value of the following under the given conditions:
c. tan(α + β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2
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