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

검증된 단계별 안내

Start by recalling the given information: \( \csc x = -3 \). Since \( \csc x = \frac{1}{\sin x} \), use this to find \( \sin x \) by taking the reciprocal, so \( \sin x = \frac{1}{\csc x} \).

Next, find \( \cos x \) using the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). Substitute the value of \( \sin x \) you found and solve for \( \cos x \). Remember that \( \cos x \) can be positive or negative depending on the quadrant where \( x \) lies.

Determine the possible quadrants for \( x \) based on the sign of \( \csc x = -3 \). Since \( \csc x \) is negative, \( \sin x \) is negative, which restricts \( x \) to quadrants III and IV. Use this to decide the sign of \( \cos x \) in each quadrant.

Recall that \( \sec x = \frac{1}{\cos x} \). Use this to rewrite the expression \( \frac{\sin x + \cos x}{\sec x} \) as \( (\sin x + \cos x) \times \cos x \).

Substitute the values of \( \sin x \) and \( \cos x \) into the expression \( (\sin x + \cos x) \times \cos x \) and simplify. Consider both possible signs of \( \cos x \) to find all possible values.

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주요 개념

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Reciprocal Trigonometric Functions

Reciprocal functions relate pairs like sine and cosecant, cosine and secant. For example, csc x = 1/sin x and sec x = 1/cos x. Knowing these relationships helps convert given values into more usable forms for solving equations.

Pythagorean Identity

The Pythagorean identity states that sin²x + cos²x = 1. This fundamental relation allows finding one trigonometric value when the other is known, essential for determining cos x when sin x is given or vice versa.

Simplifying Trigonometric Expressions

Simplifying expressions like (sin x + cos x)/sec x involves rewriting sec x as 1/cos x and then manipulating the expression algebraically. This process often reduces complex fractions to simpler forms, making it easier to substitute known values and solve.