Domanda del libro di testo
Solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x + cos x = 0
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 3, Problema 12
In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ cot θ = 3, θ lies in quadrant III.
Guida verificata passo dopo passo1
insert step 1: Understand that \( \cot \theta = 3 \) implies \( \tan \theta = \frac{1}{3} \) because \( \tan \theta = \frac{1}{\cot \theta} \).
insert step 2: Since \( \theta \) is in quadrant III, both sine and cosine are negative, but tangent is positive.
insert step 3: Use the double angle identity for tangent: \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).
insert step 4: Substitute \( \tan \theta = \frac{1}{3} \) into the double angle identity: \( \tan 2\theta = \frac{2 \times \frac{1}{3}}{1 - (\frac{1}{3})^2} \).
insert step 5: Simplify the expression to find \( \tan 2\theta \).
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Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One important identity is the double angle formula for tangent, which states that tan(2θ) = 2tan(θ) / (1 - tan²(θ)). Understanding these identities is crucial for simplifying and solving trigonometric expressions.
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Percorso guidato
5:32
Fundamental Trigonometric Identities
Quadrants and Signs of Trigonometric Functions
The unit circle is divided into four quadrants, each affecting the signs of the trigonometric functions. In quadrant III, both sine and cosine are negative, which means tangent (the ratio of sine to cosine) is positive. Knowing the quadrant in which the angle lies helps determine the signs of the trigonometric values needed for calculations.
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6:36Quadratic Formula
Finding Trigonometric Values from Cotangent
Cotangent is the reciprocal of tangent, defined as cot(θ) = 1/tan(θ). Given cot(θ) = 3, we can find tan(θ) as 1/3. This relationship allows us to derive other trigonometric values, such as sine and cosine, using the Pythagorean identity, which is essential for calculating tan(2θ) accurately.
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04:55Finding Components from Direction and Magnitude
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