Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 2.3.64

Chapter 3, Problem 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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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