Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problema 2.3.64

Capitolo 3, Problema 2.3.64

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. sin θ = 0.52991926

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Identify the given equation: \(\sin \theta = 0.52991926\). We need to find all angles \(\theta\) in the interval \([0^\circ, 360^\circ)\) that satisfy this equation.

Recall that the sine function is positive in the first and second quadrants. Therefore, there will be two solutions for \(\theta\) in the given interval: one in the first quadrant and one in the second quadrant.

Find the reference angle \(\alpha\) by taking the inverse sine (arcsin) of the given value: \(\alpha = \sin^{-1}(0.52991926)\). This will give the angle in the first quadrant.

The first solution is \(\theta_1 = \alpha\). The second solution is found by using the fact that sine is positive in the second quadrant, so \(\theta_2 = 180^\circ - \alpha\).

Express the two solutions as \(\theta_1\) and \(\theta_2\) rounded to the nearest degree, both within the interval \([0^\circ, 360^\circ)\).

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Inverse Sine Function (Arcsin)

The inverse sine function, denoted as arcsin or sin⁻¹, is used to find the angle whose sine value is given. Since sine values range between -1 and 1, arcsin returns an angle typically in the interval [-90°, 90°]. For a given sine value, arcsin provides the principal angle, which is the starting point for finding all solutions.

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Inverse Sine

Sine Function Symmetry and Periodicity

The sine function is periodic with a period of 360°, meaning sin(θ) = sin(θ + 360°k) for any integer k. Additionally, sine is positive in the first and second quadrants, so for a positive sine value, there are two angles between 0° and 360° that satisfy the equation: one in the first quadrant and one in the second quadrant, found using the reference angle.

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Period of Sine and Cosine Functions

Reference Angle and Quadrant Determination

The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. For sin θ = positive value, the reference angle is the principal angle from arcsin. The two solutions in [0°, 360°) are the reference angle itself (first quadrant) and 180° minus the reference angle (second quadrant), reflecting sine's symmetry.

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Reference Angles on the Unit Circle

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