Trigonometric Models in Real-World Scenarios?

📌 CONCEPT: Trigonometric functions can be used to model real-world scenarios, such as the orbits of celestial bodies, by describing periodic phenomena with equations involving sine, cosine, and tangent functions.

📐 RULE / FORMULA: To find the time period when the distance between the star and the planet is 250 units, we need to set the given equation 200 sin(3πt) + 250 equal to 250 and solve for t.

💡 WORKED EXAMPLE: Given the equation 200 sin(3πt) + 250 = 250, we need to isolate the sine term. Subtracting 250 from both sides gives us 200 sin(3πt) = 0. Then, dividing by 200 yields sin(3πt) = 0. Since sin(θ) = 0 at θ = 0, π, 2π, 3π, ..., we can write 3πt = 0, π, 2π, 3π, .... Solving for t gives us t = 0, 1/3, 2/3, 1, .... The time period when the distance is 250 units is t = 0, 1/3, 2/3, 1, ....

⚠️ COMMON MISTAKE: Students might forget to consider the periodic nature of trigonometric functions and solve the equation for a single value of t instead of considering all possible values.