Ch. 2 - Acute Angles and Right Triangles

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Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problema 64

Capitolo 3, Problema 64

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

Guida verificata passo dopo passo

Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). So, the equation \(\sec \theta = -\sqrt{2}\) can be rewritten as \(\frac{1}{\cos \theta} = -\sqrt{2}\).

Solve for \(\cos \theta\) by taking the reciprocal of both sides: \(\cos \theta = -\frac{1}{\sqrt{2}}\).

Recognize that \(\cos \theta = -\frac{1}{\sqrt{2}}\) is equivalent to \(\cos \theta = -\frac{\sqrt{2}}{2}\) after rationalizing the denominator.

Determine the reference angle where \(\cos \theta = \frac{\sqrt{2}}{2}\). This reference angle is \(45^\circ\) because \(\cos 45^\circ = \frac{\sqrt{2}}{2}\).

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 \(\theta = 180^\circ + 45^\circ\).

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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.

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Graphs of Secant and Cosecant Functions

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.

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How to Solve Linear Trigonometric Equations

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.

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Quadratic Formula

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