Domanda del libro di testo
In Exercises 7–14, use the given information to find the exact value of each of the following:
b. cos 2θ
15
sin θ = -------- , θ lies in quadrant II.
17
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 3, Problema 7
Use substitution to determine whether the given x-value is a solution of the equation.
Guida verificata passo dopo passo1
First, substitute the given value of \(x = \frac{5\pi}{12}\) into the left side of the equation \(\tan 2x\). This means calculating \(\tan\left(2 \times \frac{5\pi}{12}\right)\).
Simplify the expression inside the tangent function: \(2 \times \frac{5\pi}{12} = \frac{10\pi}{12} = \frac{5\pi}{6}\), so you need to find \(\tan\left(\frac{5\pi}{6}\right)\).
Recall or use the unit circle to find the exact value of \(\tan\left(\frac{5\pi}{6}\right)\). Remember that \(\frac{5\pi}{6}\) is in the second quadrant where tangent is negative.
Next, evaluate the right side of the equation, which is \(-\frac{\sqrt{3}}{3}\). This is a constant value you can compare with the left side.
Finally, compare the value of \(\tan\left(\frac{5\pi}{6}\right)\) with \(-\frac{\sqrt{3}}{3}\). If they are equal, then \(x = \frac{5\pi}{12}\) is a solution; if not, it is not a solution.
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Tangent Function and Its Properties
The tangent function, tan(θ), is defined as the ratio of sine to cosine (sin θ / cos θ). It is periodic with period π, meaning tan(θ + π) = tan(θ). Understanding how to evaluate tangent at specific angles, especially multiples of π, is essential for solving equations involving tan(2x).
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Percorso guidato
5:43
Introduction to Tangent Graph
Substitution Method in Trigonometric Equations
Substitution involves replacing the variable x with a given value to verify if it satisfies the equation. Here, substituting x = 5π/12 into tan(2x) allows direct evaluation to check if it equals the given expression. This method helps confirm whether the proposed x-value is a solution.
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Percorso guidato
04:42Solve Trig Equations Using Identity Substitutions
Simplifying and Evaluating Trigonometric Expressions
Evaluating tan(2x) at x = 5π/12 requires simplifying the angle 2x = 5π/6 and then calculating tan(5π/6). Knowing exact values of tangent at standard angles and simplifying radicals like √3/3 is crucial to compare both sides of the equation accurately.
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Percorso guidato
6:36Simplifying Trig Expressions
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In Exercises 1–60, verify each identity.
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Domanda del libro di testo
In Exercises 7–14, use the given information to find the exact value of each of the following:
a. sin 2θ
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sin θ = -------- , θ lies in quadrant II.
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In Exercises 1–6, use the figures to find the exact value of each trigonometric function.
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Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.
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