Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. sec θ = -√2

Verified step by step guidance

1. Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). So, the equation \(\sec \theta = -\sqrt{2}\) can be rewritten as \(\frac{1}{\cos \theta} = -\sqrt{2}\).
2. Solve for \(\cos \theta\) by taking the reciprocal of both sides: \(\cos \theta = -\frac{1}{\sqrt{2}}\).
3. Recognize that \(\cos \theta = -\frac{1}{\sqrt{2}}\) is equivalent to \(\cos \theta = -\frac{\sqrt{2}}{2}\) after rationalizing the denominator.
4. Determine the reference angle where \(\cos \theta = \frac{\sqrt{2}}{2}\). This reference angle is \(45^\circ\) because \(\cos 45^\circ = \frac{\sqrt{2}}{2}\).
5. Since \(\cos \theta\) is negative, find all angles in the interval \([0^\circ, 360^\circ)\) where cosine is negative. Cosine is negative in the second and third quadrants, so the solutions are \(\theta = 180^\circ - 45^\circ\) and \(\theta = 180^\circ + 45^\circ\).

Key Concepts

Definition of Secant Function

The secant function, sec θ, is the reciprocal of the cosine function, defined as sec θ = 1/cos θ. Understanding this relationship allows us to convert secant equations into cosine equations, which are often easier to solve.

Solving Trigonometric Equations in a Given Interval

When solving trigonometric equations like sec θ = -√2 over [0°, 360°), it is essential to find all angles θ within the interval that satisfy the equation. This involves considering the periodicity and sign of the trigonometric functions in different quadrants.

Sign of Trigonometric Functions in Quadrants

The sign of cosine (and thus secant) varies by quadrant: cosine is positive in the first and fourth quadrants and negative in the second and third. Since sec θ = 1/cos θ, secant shares the same sign pattern, which helps identify the correct quadrants for solutions.