The graph of y-7 cosθ + 24 sinθ would be a sinusoid if it were plotted. What would be the first positive value of θ at which there is a high point?
From the graph, I get about θ=1.3
I assume you mean
y = 7 cos θ + 24 sin θ
well, my inclination is to look for where the derivative is 0
dy/dθ = -7 sin θ + 24 cos θ = 0
or
tan θ = 24/7 = 3.43
θ = 73.7 degrees
yes, about 1.3 radians
To find the first positive value of θ at which there is a high point in the graph of y = -7cosθ + 24sinθ, we need to find the maximum value of y.
A sinusoid with the equation y = A cosθ + B sinθ can be rewritten using trigonometric identities as:
y = √(A² + B²) sin(θ + φ),
where φ is the phase shift angle given by φ = tan⁻¹(B/A).
In our case, A = -7 and B = 24, so
φ = tan⁻¹(B/A) = tan⁻¹(24/(-7)) ≈ -1.266 rad.
The maximum value of the sinusoid is given by the amplitude, which is the square root of the sum of the squares of A and B. In this case, the amplitude is √((-7)² + 24²) = 25.
The graph of the sinusoid will have high points when the value of sin(θ + φ) is 1. So we need to find the first positive value of θ that satisfies sin(θ + φ) = 1.
sin(θ + φ) = 1 when (θ + φ) = π/2 + 2πn, or θ = (π/2 + 2πn) - φ, where n is an integer.
Substituting the values of φ and solving for θ, we get:
θ = (π/2 + 2πn) - (-1.266) = π/2 + 2πn + 1.266.
Since we are looking for the first positive value of θ, we substitute n = 0:
θ = π/2 + 2π(0) + 1.266 ≈ 2.038 rad.
Therefore, the first positive value of θ at which there is a high point in the graph is approximately 2.038 radians.