Imagine you need to calculate the height of a tower. By observing its shadow and measuring the distance between you and the base of the tower, you discover that the angle formed between your vertical line and the line connecting you to the base of the tower is 60 degrees. Additionally, the distance between your vertical line and the base of the tower is 40 meters. To find the height of the tower, you need to solve the following trigonometric equation: h = d * tan(θ), where h is the height of the tower, d is the horizontal distance between the observer's vertical line and the base of the tower, and θ is the angle of inclination. What is the height of the tower?

Given that $$x$$ is equal to the cosine of 2 radians and $$y$$ is equal to the sine of 2 radians, it is concluded that

 is equal to the cosine of 2 radians and 

 is equal to the sine of 2 radians, it is concluded that

If x ∈ [0,2π], the number of solutions of the equation 2sin^3 x - sin x + 1 = cos^2 x is equal to:

To solve the equation 2sin^3 x - sin x + 1 = cos^2 x, follow the steps below: Step 1: Use the trigonometric identity sin^2 x + cos^2 x = 1. Replace cos^2 x with (1 - sin^2 x) in the original equation: 2sin^3 x - sin x + 1 = 1 - sin^2 x Step 2: Rearrange the equation. 2sin^3 x - sin x - sin^2 x + sin^2 x = 0 Now, we have a cubic equation in sin x: 2sin^3 x - sin x = 0 Step 3: Factor the equation. We can factor out "sin x" from the equation: sin x(2sin^2 x - 1) = 0 Step 4: Find the solutions for the equation. Now we have two equations to solve: 1) sin x = 0 2) 2sin^2 x - 1 = 0 For equation 1, the solutions are: x = 0, x = π For equation 2, first solve for sin^2 x: sin^2 x = 1/2 Now, find sin x: sin x = ±√(1/2) sin x = ±(1/√2) The solutions for these sin x values in the interval [0, 2π] are: x = π/4, x = 3π/4, x = 5π/4, x = 7π/4 Step 5: Combine all solutions. Combining all solutions, we have: x = 0, x = π/4, x = 3π/4, x = π, x = 5π/4, x = 7π/4 Therefore, there are a total of 6 possible solutions for the equation in the interval [0, 2π]. The correct answer is "6".

During a bird watching activity, you want to calculate the height of a tree without climbing it. Using an instrument that measures the angle in relation to the ground, you determine that the angle of elevation from the top of the tree from the point where you are is 60 degrees. If you are 15 meters away from the base of the tree and continue in a straight line, calculate the height of the tree. Use the trigonometric equation to solve this problem.

In a physics experiment, a student observes that the shadow of a swinging pendulum projects a movement that can be modeled by the trigonometric function y = 5cos(2πt + π/2), where y is the horizontal distance of the pendulum's shadow from a fixed point and t is the time in seconds. At what time instant t after the start of the experiment will the pendulum's shadow be 5 meters away from the fixed point for the first time?

Question bank: Trigonometric Equation

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