Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.

Verified step by step guidance

1. Recognize that \( \csc x = \frac{1}{\sin x} \), so if \( \csc x = -3 \), then \( \sin x = -\frac{1}{3} \).
2. Use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to find \( \cos x \). Substitute \( \sin x = -\frac{1}{3} \) into the identity to get \( \left(-\frac{1}{3}\right)^2 + \cos^2 x = 1 \).
3. Solve for \( \cos^2 x \) to find \( \cos x \). This gives \( \cos^2 x = 1 - \frac{1}{9} \), so \( \cos^2 x = \frac{8}{9} \). Therefore, \( \cos x = \pm \frac{2\sqrt{2}}{3} \).
4. Calculate \( \sec x \) using \( \sec x = \frac{1}{\cos x} \). For \( \cos x = \frac{2\sqrt{2}}{3} \), \( \sec x = \frac{3}{2\sqrt{2}} \), and for \( \cos x = -\frac{2\sqrt{2}}{3} \), \( \sec x = -\frac{3}{2\sqrt{2}} \).
5. Substitute \( \sin x \), \( \cos x \), and \( \sec x \) into the expression \( \frac{\sin x + \cos x}{\sec x} \) and simplify for both cases of \( \cos x \).

Key Concepts

Cosecant and Its Relationship to Sine

Cosecant (csc) is the reciprocal of sine, defined as csc x = 1/sin x. If csc x = -3, it implies that sin x = -1/3. Understanding this relationship is crucial for solving the problem, as it allows us to find the sine value needed for further calculations.

Secant and Its Relationship to Cosine

Secant (sec) is the reciprocal of cosine, defined as sec x = 1/cos x. To find the expression (sin x + cos x)/sec x, we need to determine the cosine value. This requires using the Pythagorean identity sin² x + cos² x = 1 to find cos x based on the known value of sin x.

Trigonometric Identities and Simplification

Trigonometric identities, such as the Pythagorean identity and reciprocal identities, are essential for simplifying expressions involving trigonometric functions. In this case, after finding sin x and cos x, we can substitute these values into the expression (sin x + cos x)/sec x and simplify it to find the final result.