Ch. 3 - Trigonometric Identities and Equations

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Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problema 3.RE.35b

Capitolo 3, Problema 3.RE.35b

In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

Guida verificata passo dopo passo

Identify the given information: \( \sin \alpha = \frac{3}{5} \) with \( 0 < \alpha < \frac{\pi}{2} \), and \( \sin \beta = \frac{12}{13} \) with \( \frac{\pi}{2} < \beta < \pi \).

Determine the quadrant of each angle to find the signs of \( \cos \alpha \) and \( \cos \beta \). Since \( \alpha \) is in the first quadrant, \( \cos \alpha > 0 \). Since \( \beta \) is in the second quadrant, \( \cos \beta < 0 \).

Use the Pythagorean identity to find \( \cos \alpha \) and \( \cos \beta \): \[ \cos \theta = \pm \sqrt{1 - \sin^2 \theta} \] Calculate: \[ \cos \alpha = +\sqrt{1 - \left(\frac{3}{5}\right)^2} \] \[ \cos \beta = -\sqrt{1 - \left(\frac{12}{13}\right)^2} \]

Recall the cosine difference identity: \[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]

Substitute the values of \( \sin \alpha \), \( \sin \beta \), \( \cos \alpha \), and \( \cos \beta \) into the identity to express \( \cos(\alpha - \beta) \) exactly.

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Sum and Difference Formulas for Cosine

The cosine of the difference of two angles, cos(α - β), can be found using the formula cos(α - β) = cos α cos β + sin α sin β. This identity allows us to express the cosine of a difference in terms of the sines and cosines of the individual angles, which is essential when given sine values and angle ranges.

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Percorso guidato

2:25

Verifying Identities with Sum and Difference Formulas

Determining Cosine from Sine and Quadrant Information

Given sin α and sin β along with their angle ranges, we can find cos α and cos β using the Pythagorean identity cos²θ = 1 - sin²θ. The sign of cosine depends on the quadrant of the angle, so knowing the interval for α and β helps determine whether cosine is positive or negative.

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Percorso guidato

5:08

Sine, Cosine, & Tangent of 30°, 45°, & 60°

Angle Measurement in Radians and Quadrant Boundaries

Angles are given in radians with specified intervals (e.g., 0 < α < 3π/2). Understanding these intervals helps identify the quadrant in which each angle lies, which is crucial for determining the signs of trigonometric functions and correctly applying identities.

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Percorso guidato

5:04

Converting between Degrees & Radians

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