Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable.
cos θ = ―5/8 , and θ is in quadrant III

Verified step by step guidance

1. Recall that the six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). We are given cos \( \theta = -\frac{5}{8} \) and that \( \theta \) is in quadrant III.
2. Since \( \theta \) is in quadrant III, both sine and cosine are negative, and tangent is positive. Use the Pythagorean identity to find sin \( \theta \): \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substitute \( \cos \theta = -\frac{5}{8} \): \[ \sin^2 \theta + \left(-\frac{5}{8}\right)^2 = 1 \] Simplify and solve for \( \sin^2 \theta \).
3. Take the square root of \( \sin^2 \theta \) to find \( \sin \theta \). Since \( \theta \) is in quadrant III, \( \sin \theta \) is negative.
4. Find \( \tan \theta \) using the definition: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] Use the values of \( \sin \theta \) and \( \cos \theta \) found in previous steps.
5. Calculate the reciprocal functions: \[ \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta} \] Remember to rationalize denominators if necessary.

Key Concepts

Trigonometric Functions and Their Relationships

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are interrelated ratios based on a right triangle or the unit circle. Knowing one function value, such as cosine, allows calculation of others using identities like sin²θ + cos²θ = 1 and definitions like tanθ = sinθ/cosθ.

Sign of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant of the angle. In quadrant III, both sine and cosine are negative, while tangent is positive. This knowledge helps determine the correct signs of all six functions when given one value and the quadrant.

Rationalizing Denominators

Rationalizing denominators involves eliminating radicals from the denominator of a fraction by multiplying numerator and denominator by a suitable expression. This is often required for final answers in trigonometry to present values in a simplified, standardized form.