Domanda del libro di testo
In Exercises 53–62, solve each equation on the interval [0, 2𝝅). cot x (tan x - 1) = 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 3.3.53
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.3.53Capitolo 3, Problema 3.3.53
In Exercises 47–54, use the figures to find the exact value of each trigonometric function. 2sin(θ/2)cos(θ/2)
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Identify the given trigonometric expressions: \( 2 \sin \frac{\theta}{2} \) and \( \cos \frac{\theta}{2} \). The problem likely involves finding exact values related to these half-angle expressions.
Recall the half-angle formulas for sine and cosine:
\[ \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \quad \text{and} \quad \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \]
Determine the correct sign based on the quadrant where \( \frac{\theta}{2} \) lies.
Substitute the known value of \( \cos \theta \) from the figure or given information into the half-angle formulas to express \( \sin \frac{\theta}{2} \) and \( \cos \frac{\theta}{2} \) in exact radical form.
Multiply \( \sin \frac{\theta}{2} \) by 2 as indicated in the expression \( 2 \sin \frac{\theta}{2} \) to simplify or rewrite the expression accordingly.
Combine the expressions or use Pythagorean identities if needed to find the exact values of the trigonometric functions requested, ensuring all steps use exact values (no decimal approximations).
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Trigonometric Functions and Their Definitions
Trigonometric functions like sine and cosine relate the angles of a triangle to the ratios of its sides. Specifically, sine of an angle is the ratio of the opposite side to the hypotenuse, while cosine is the ratio of the adjacent side to the hypotenuse. Understanding these definitions is essential for evaluating trigonometric expressions.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Angle Halving and the Use of θ/2
The expression involves trigonometric functions of half an angle (θ/2). Recognizing how to work with half-angle values is important, as it often requires applying half-angle identities or understanding how the angle relates to the original θ to find exact values.
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Percorso guidato
04:42Solve Trig Equations Using Identity Substitutions
Half-Angle Identities
Half-angle identities provide formulas to find sine and cosine of half an angle using the cosine or sine of the original angle θ. For example, sin(θ/2) = ±√((1 - cos θ)/2) and cos(θ/2) = ±√((1 + cos θ)/2). These identities are crucial for finding exact trigonometric values when given θ.
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Percorso guidato
05:06Double Angle Identities
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Domanda del libro di testo
In Exercises 53–62, solve each equation on the interval [0, 2𝝅). sin x + 2 sin x cos x = 0
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Domanda del libro di testo
In Exercises 1–6, use the figures to find the exact value of each trigonometric function.
cos 2θ
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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 cos² x + 3 cos x + 1 = 0
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Domanda del libro di testo
In Exercises 59–68, verify each identity.
cos²(θ/2) = (sec θ + 1)/(2 sec θ)
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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. 6 sin⁴ x
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