Find the exact value of each real number y if it exists. Do not use a calculator.
y = sec⁻¹ 1

Verified step by step guidance

Recall the definition of the inverse secant function: \(y = \sec^{-1}(x)\) means \(\sec(y) = x\) and \(y\) lies in the principal range of \(\sec^{-1}\), which is \([0, \pi]\) excluding \(\frac{\pi}{2}\).

Set up the equation from the problem: \(\sec(y) = 1\).

Recall that \(\sec(y) = \frac{1}{\cos(y)}\), so the equation becomes \(\frac{1}{\cos(y)} = 1\).

Solve for \(\cos(y)\): multiply both sides by \(\cos(y)\) and divide both sides by 1 (which does not change the equation), giving \(\cos(y) = 1\).

Find all \(y\) in the principal range \([0, \pi]\) (excluding \(\frac{\pi}{2}\)) such that \(\cos(y) = 1\). Identify these values as the solutions for \(y\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Secant Function (sec⁻¹ x)

The inverse secant function, sec⁻¹ x, returns the angle whose secant is x. It is defined for |x| ≥ 1, and its range is typically [0, π] excluding π/2. Understanding this function helps find the angle y such that sec y = x.

Secant Function Definition

The secant function, sec θ, is the reciprocal of the cosine function: sec θ = 1/cos θ. Knowing this relationship allows you to rewrite sec⁻¹ 1 as finding an angle where cos θ = 1, simplifying the problem.

Exact Values of Trigonometric Functions

Certain angles have well-known exact trigonometric values, such as cos 0 = 1. Recognizing these standard values enables solving inverse trig problems without a calculator by matching the given value to a known angle.

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