Berechnung**: \(1.33 \times \sin(30°) = 1.5 \times \sin(\theta_2)\)
["Title: How to Solve the Triangle: Solving (1.33 \ imes \sin(30^\circ) = 1.5 \ imes \sin(\ heta_2))", "---", "Introduction", "In trigonometry, solving for unknown angles in triangles often involves simplifying equations using known values and fundamental identities. A common type of problem students encounter is when sine values are set equal so that the unknowns can be compared using inverse sine. In this article, we’ll explore how to solve the equation:", "[\n1.33 \ imes \sin(30^\circ) = 1.5 \ imes \sin(\ heta_2)\n]", "We’ll break down the steps clearly and explain the key trigonometric principles involved, providing a practical example suitable for students, educators, and self-learners.", "---", "### Step-by-Step Calculation", "Step 1: Evaluate known sine value", "We begin with:", "[\n\sin(30^\circ) = 0.5\n]", "So, substitute this into the left-hand side:", "[\n1.33 \ imes 0.5 = 0.665\n]", "Now the equation becomes:", "[\n0.665 = 1.5 \ imes \sin(\ heta_2)\n]", "---", "Step 2: Solve for (\sin(\ heta_2))", "Isolate (\sin(\ heta_2)) by dividing both sides by 1.5:", "[\n\sin(\ heta_2) = \frac{0.665}{1.5} = 0.4433\overline{3}\n]", "---", "Step 3: Find (\ heta_2) using the inverse sine function", "Use the inverse sine (arcsin) function:", "[\n\ heta_2 = \arcsin(0.4433\overline{3})\n]", "Calculating this using a calculator (in degree mode):", "[\n\ heta_2 \approx 26.26^\circ\n]", "However, note that the sine function is positive in the first and second quadrants. Hence, the second possible solution is:", "[\n\ heta_2 = 180^\circ - 26.26^\circ = 153.74^\circ\n]", "But since (\ heta_2) usually represents an angle in a triangle (acute by definition), the valid solution in standard triangular contexts is:", "[\n\ heta_2 \approx 26.3^\circ\n]", "---", "### Understanding the Problem in Context", "This equation models a triangle scenario where two sine terms are proportional—common in problems involving height, distance, and angles in construction, physics, or surveying. By simplifying the sine values, we isolate the unknown angle and apply inverse trigonometric functions effectively.", "---", "### Final Answer", "[\n\boxed{\ heta_2 \approx 26.3^\circ}\n]\n(since (\ heta_2) lies in the first quadrant and represents a valid angle in a triangle)", "---", "### Tips for Solving Similar Problems", "- Know key sine values: Memorize (\sin(30^\circ) = 0.5), (\sin(45^\circ) \approx 0.707), (\sin(60^\circ) \approx 0.866).\n- Use identities wisely: When proportionality occurs, plug in sine values directly.\n- Consider the quadrant: (\sin(\ heta) = k) yields two angles unless restricted; usually, only the acute angle applies to triangles.\n- Apply calculator carefully: Always use degree mode when solving triangle angles.", "---", "### Summary", "Solving equations like (1.33 \ imes \sin(30^\circ) = 1.5 \ imes \sin(\ heta_2)) involves calculating known sine values, isolating the unknown trigonometric ratio, and applying (\arcsin). The primary solution in trigonometric triangle problems typically falls in the first quadrant, typically around (26.3^\circ) in this case. Understanding these steps builds a solid foundation for tackling more complex trigonometric relationships in geometry and applied mathematics.", "---", "Keywords:\nberechnung sin(30°), trigonometrische Gleichung lösen, inverse Sinus berechnen, sinθ-Rechnung, Dreiecksberechnung, trigonometry example, solving sine equation."]
