Question

Please complete all questions

1.
Verify the basic identity. What is the domain of validity? Cot θ = cos θ csc θ

2.
Verify the identity. Justify each step. tan θ+cot θ= (1 / sin θ cos θ)

3.
A weight is attached to a spring that is fixed to the floor. The equation h=7cos (π/3 t) models the height, h, in centimeters after t seconds of the weight being stretched and released.
a. Solve the equation for t.

b. Find the times at which the weight is first at a height of 1 cm, of 3 cm, and of 5 cm above the rest position. Calculate your answer using radian angle measure. Round your answers to the nearest hundredth.

Answer 1

[AI-generated answer — Bot GPT 3.5]

1. The basic identity is Cot θ = cos θ csc θ, where the domain of validity is all values of θ except where sin θ = 0.

2. tan θ + cot θ = (1 / sin θ cos θ)
= (sin θ / cos θ) + (cos θ / sin θ)
= (sin^2 θ + cos^2 θ) / (sin θ cos θ)
= 1 / (sin θ cos θ)

3.
a. To solve the equation h = 7cos(π/3 t) for t, we need to isolate t.
Divide both sides by 7: cos(π/3 t) = h/7
Take the inverse cosine of both sides: π/3 t = arccos(h/7)
Multiply by 3/π: t = 3/π * arccos(h/7)

b.
- At a height of 1 cm:
t = 3/π * arccos(1/7) ≈ 1.16 seconds

- At a height of 3 cm:
t = 3/π * arccos(3/7) ≈ 1.87 seconds

- At a height of 5 cm:
t = 3/π * arccos(5/7) ≈ 2.31 seconds