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.35e
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.RE.35eCapitolo 3, Problema 3.RE.35e
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.
Guida verificata passo dopo passo1
Identify the given information: \( \sin \alpha = \frac{3}{5} \) with \( 0 < \alpha < \frac{\pi}{2} \), and \( \sin \beta = \frac{12}{13} \) with \( \frac{\pi}{2} < \beta < \pi \).
Since \( \alpha \) is in the first quadrant (between 0 and \( \frac{\pi}{2} \)), both \( \sin \alpha \) and \( \cos \alpha \) are positive. Use the Pythagorean identity to find \( \cos \alpha \):
\(\cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left(\frac{3}{5}\right)^2}\)
Since \( \beta \) is in the second quadrant (between \( \frac{\pi}{2} \) and \( \pi \)), \( \sin \beta \) is positive but \( \cos \beta \) is negative. Use the Pythagorean identity to find \( \cos \beta \):
\(\cos \beta = -\sqrt{1 - \sin^2 \beta} = -\sqrt{1 - \left(\frac{12}{13}\right)^2}\)
Use the cosine of difference formula to find \( \cos(\beta - \alpha) \):
\(\cos(\beta - \alpha) = \cos \beta \cos \alpha + \sin \beta \sin \alpha\)
Substitute the values of \( \cos \alpha \), \( \cos \beta \), \( \sin \alpha \), and \( \sin \beta \) into the formula and simplify to find the exact value of \( \cos(\beta - \alpha) \).
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Trigonometric Ratios and Their Definitions
Trigonometric ratios like sine and cosine relate the angles of a triangle to the ratios of its sides. Understanding how to interpret and use these ratios is essential for finding exact values of trigonometric functions given certain angle conditions.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Quadrant and Angle Restrictions
The given angle restrictions (e.g., 0 < α < π/2 and π/2 < β < π) determine the sign and possible values of trigonometric functions. Knowing which quadrant an angle lies in helps identify whether sine, cosine, or other functions are positive or negative.
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6:36Quadratic Formula
Using Pythagorean Identity to Find Unknown Values
The Pythagorean identity, sin²θ + cos²θ = 1, allows calculation of one trigonometric function when the other is known. This is crucial for finding exact values of cosine or sine when only one ratio and the angle’s quadrant are given.
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6:25Pythagorean Identities
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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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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
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Domanda del libro di testo
In Exercises 55–58, use the given information to find the exact value of each of the following:
a. sin(α/2)
sec α = ﹣3, 𝝅/2 < α < 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:
a. sin(α + β)
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 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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