Question

Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to two decimal places where appropriate.)

cos(θ) = 0.72
θ =

List six specific solutions.
θ =

Answer 1

θ = 0.77+6.28k, -0.77+6.28k, includes {-7.05, -5.52, -0.77, 0.77, 5.52, 7.05}

Explanation:

The principal value of θ can be found using the inverse cosine function:

θ = arccos(0.72) ≈ 0.766994

Additional values of θ that also satisfy the equation are values that are this far away from any multiple of 2π, that is, ...

θ = ±arccos(0.72) + 2kπ ≈ ±0.77 +6.28k

Some approximate values for θ are {-7.05, -5.52, -0.77, 0.77, 5.52, 7.05}.

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Comment on multiples of π

Since π is irrational, the larger the multiple, the larger the error for any given approximation of π. One needs to be careful to use a suitable approximation for the multiple of interest. The same problem affects calculator accuracy for angles of very large magnitude.