\times \sin(30^\circ) = 1.50 \times \sin(\theta_2)
["# Solving the Trigonometric Equation: (\sin(30^\circ) = 1.50 \ imes \sin(\ heta_2))", "Understanding trigonometric equations is fundamental for students and professionals in mathematics, physics, engineering, and related fields. One such equation that appears in various applications—from wave mechanics to navigation—is:", "[\n\sin(30^\circ) = 1.50 \ imes \sin(\ heta_2)\n]", "In this article, we’ll break down how to solve this equation step-by-step and explore its mathematical significance.", "---", "## What is the Equation Trying to Solve?", "The equation relates a known sine value to an unknown angle (\ heta_2) through a scaling factor of 1.50. Since (\sin(30^\circ) = 0.5), we rewrite the equation as:", "[\n0.5 = 1.50 \ imes \sin(\ heta_2)\n]", "Our goal is to find (\ heta_2) such that this identity holds true.", "---", "## Step-by-Step Solution", "### Step 1: Isolate (\sin(\ heta_2))", "Start by dividing both sides by 1.50:", "[\n\sin(\ heta_2) = \frac{0.5}{1.50} = \frac{1}{3} \approx 0.3333\n]", "So:", "[\n\sin(\ heta_2) = \frac{1}{3}\n]", "### Step 2: Solve for (\ heta_2)", "To find (\ heta_2), take the inverse sine (arcsine) of both sides:", "[\n\ heta_2 = \arcsin\left(\frac{1}{3}\right)\n]", "Using a calculator (in degree mode):", "[\n\ heta_2 \approx 19.47^\circ\n]", "### Step 3: Account for All Possible Solutions", "The sine function is positive in two quadrants: Quadrant I and Quadrant II. Therefore, the general solutions are:", "- Primary solution:\n [\n \ heta_2 = \arcsin\left(\frac{1}{3}\right) \approx 19.47^\circ\n ]", "- Supplementary solution:\n [\n \ heta_2 = 180^\circ - 19.47^\circ = 160.53^\circ\n ]", "So all solutions within (0^\circ \leq \ heta_2 < 360^\circ) are approximately:\n[\n\ heta_2 \approx 19.47^\circ \quad \ ext{and} \quad \ heta_2 \approx 160.53^\circ\n]", "---", "## Why Is This Equation Important?", "This type of equation often arises in problems involving:", "- Oscillations and waves: Determining phase angles in sinusoidal signals.\n- Projectile motion: Resolving vertical components of motion.\n- Navigation and surveying: Calculating bearing angles from trigonometric relationships.\n- Physics applications: Finding angles in right triangles with known ratios.", "---", "## Practical Tips for Solving Trigonometric Equations", "- Always recall exact values (e.g., (\sin(30^\circ) = 0.5)).\n- Use a calculator in degree mode for accurate (\arcsin) results.\n- Remember the periodicity and symmetry of sine function to find all valid solutions.\n- Verify solutions by substituting back into the original equation.", "---", "## Summary", "To solve:", "[\n\sin(30^\circ) = 1.50 \ imes \sin(\ heta_2)\n]", "we found:", "[\n\sin(\ heta_2) = \frac{1}{3}, \quad \ heta_2 \approx 19.47^\circ \ ext{ or } 160.53^\circ\n]", "This approach applies broadly to trigonometric problem-solving and underpins numerous scientific and engineering calculations.", "---", "## Want More?", "Explore related topics like solving (\sin(\ heta) = k), solving trigonometric equations graphically, or using the sine law in triangle solving. Mastery of such equations increases confidence in handling more complex real-world science and math challenges.", "---", "Keywords: (\sin(30^\circ) = 1.50 \ imes \sin(\ heta_2)), solve trigonometric equation, sine identities, inverse sine, (\arcsin), chemistry/physics applications, angle calculation, trigonometry tutorial."]
