Find the remaining five trigonometric functions of θ.
csc θ = -5/2, θ in quadrant III

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Recall that the cosecant function is the reciprocal of the sine function, so we can find \(\sin \theta\) by taking the reciprocal of \(\csc \theta\). Write this as \(\sin \theta = \frac{1}{\csc \theta}\).

Calculate \(\sin \theta\) using the given value \(\csc \theta = -\frac{5}{2}\), so \(\sin \theta = -\frac{2}{5}\). Remember the sign is negative because \(\csc \theta\) is negative in quadrant III.

Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos \theta\). Substitute \(\sin \theta = -\frac{2}{5}\) and solve for \(\cos \theta\).

Determine the sign of \(\cos \theta\) in quadrant III. Since cosine is negative in quadrant III, choose the negative root for \(\cos \theta\).

Find the remaining trigonometric functions using the definitions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Use the values of \(\sin \theta\) and \(\cos \theta\) found in previous steps.

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Reciprocal Trigonometric Functions

Each trigonometric function has a reciprocal counterpart: sine and cosecant, cosine and secant, tangent and cotangent. Knowing one function allows you to find its reciprocal by taking the inverse (e.g., csc θ = 1/sin θ). This relationship is essential for determining missing functions when one is given.

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Introduction to Trigonometric Functions

Sign of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant of the angle θ. In quadrant III, both sine and cosine are negative, while tangent is positive. Understanding these sign rules helps correctly assign positive or negative values to the functions when calculating them.

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Pythagorean Identity

The Pythagorean identity, sin²θ + cos²θ = 1, links sine and cosine values. Given one function, you can use this identity to find the other. This is crucial for finding all trigonometric functions when only one is known, especially when combined with quadrant sign information.

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