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Bot GPT 3.5 1.
- To verify the basic identity Cot θ = cos θ csc θ:
- Recall that Cot θ = cos θ / sin θ and csc θ = 1 / sin θ
- Therefore, Cot θ = cos θ * (1 / sin θ)
- Simplifying, we get Cot θ = cos θ csc θ
- The domain of validity for this identity is where sin θ is not equal to 0 (since division by 0 is undefined), so the domain is all real numbers except when sin θ = 0.
2.
- To verify the identity tan θ + cot θ = 1 / (sin θ cos θ):
- Recall that tan θ = sin θ / cos θ
- Substitute tan θ = sin θ / cos θ into the expression: sin θ / cos θ + cos θ / sin θ
- Find a common denominator: (sin^2 θ + cos^2 θ) / (sin θ cos θ)
- Recall the Pythagorean identity: sin^2 θ + cos^2 θ = 1
- Therefore, (1) / (sin θ cos θ) = 1 / (sin θ cos θ)
- Justification for each step is provided by showing the mathematical operations performed and simplifications made.
3.
a.
- To solve the equation h = 7cos (π/3 t) for t:
- Rewrite the equation as 7cos (π/3 t) = h
- Divide both sides by 7 to isolate the cosine term: cos (π/3 t) = h/7
- Take the arccosine of both sides: π/3 t = arccos(h/7)
- Solve for t: t = (3/arccos(h/7)) * arccos(h/7)
b.
- To find the times at which the weight is first at a height of 1 cm, 3 cm, and 5 cm above the rest position:
- Substitute h = 1 cm, 3 cm, and 5 cm into the equation for t found in part (a)
- Calculate the values of t using radian angle measure and rounding to the nearest hundredth of a second.