Find sinθ.
csc θ = -8/5

Verified step by step guidance

1. Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\). This means \(\sin \theta = \frac{1}{\csc \theta}\).
2. Substitute the given value of \(\csc \theta = -\frac{8}{5}\) into the equation: \(\sin \theta = \frac{1}{-\frac{8}{5}}\).
3. Simplify the fraction by multiplying numerator and denominator: \(\sin \theta = -\frac{5}{8}\).
4. Consider the sign of \(\sin \theta\) based on the quadrant where \(\theta\) lies. Since \(\csc \theta\) is negative, \(\sin \theta\) is also negative, which matches the simplified value.
5. Conclude that \(\sin \theta = -\frac{5}{8}\), consistent with the given \(\csc \theta\).

Key Concepts

Reciprocal Trigonometric Functions

The cosecant function (csc θ) is the reciprocal of the sine function (sin θ), meaning csc θ = 1/sin θ. To find sin θ when given csc θ, you take the reciprocal of the given value.

Sign of Trigonometric Functions in Quadrants

The sign of sine and cosecant depends on the quadrant where the angle θ lies. Since csc θ = -8/5 is negative, sin θ must also be negative, indicating θ is in either the third or fourth quadrant.

Simplifying and Interpreting Fractional Values

Given csc θ = -8/5, sin θ is the reciprocal, so sin θ = -5/8. Understanding how to simplify and interpret these fractional values is essential for accurate calculation and further problem-solving.