Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. sec θ = -2√3 3

Verified step by step guidance

1. Recall that the secant function is the reciprocal of the cosine function, so we can write the equation as \(\sec \theta = -\frac{2\sqrt{3}}{3}\), which implies \(\cos \theta = -\frac{3}{2\sqrt{3}}\) after taking the reciprocal.
2. Simplify the expression for \(\cos \theta\) by rationalizing the denominator if necessary, to get it into a more recognizable form.
3. Determine the reference angle by finding the angle whose cosine is the positive value of the simplified expression, using the inverse cosine function: \(\theta_{ref} = \cos^{-1}(\text{positive value})\).
4. Since \(\cos \theta\) is negative, identify the quadrants where cosine is negative. Cosine is negative in Quadrants II and III, so the solutions for \(\theta\) will be in these quadrants.
5. Find the actual angles in the interval \([0^\circ, 360^\circ)\) by using the reference angle and the knowledge of cosine signs in each quadrant: \(\theta = 180^\circ - \theta_{ref}\) (Quadrant II) and \(\theta = 180^\circ + \theta_{ref}\) (Quadrant III).

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 values into cosine values, which are often easier to work with when solving trigonometric equations.

Solving Trigonometric Equations in a Given Interval

When solving for θ in a specific interval, such as [0°, 360°), it is important to find all angles that satisfy the equation within that range. This involves considering the signs and values 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.