Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable.
sec θ = ―√5 , and θ is in quadrant II

Verified step by step guidance

1. Recall the definition of the secant function: \(\sec \theta = \frac{1}{\cos \theta}\). Given \(\sec \theta = -\sqrt{5}\), find \(\cos \theta\) by taking the reciprocal: \(\cos \theta = \frac{1}{\sec \theta} = \frac{1}{-\sqrt{5}}\).
2. Rationalize the denominator of \(\cos \theta\): multiply numerator and denominator by \(\sqrt{5}\) to get \(\cos \theta = -\frac{\sqrt{5}}{5}\).
3. Since \(\theta\) is in quadrant II, recall the signs of trigonometric functions there: \(\cos \theta\) is negative, \(\sin \theta\) is positive. This confirms the sign of \(\cos \theta\) is correct.
4. Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\sin \theta\). Substitute \(\cos \theta = -\frac{\sqrt{5}}{5}\) and solve for \(\sin \theta\).
5. Once \(\sin \theta\) is found, determine the remaining trigonometric functions using their definitions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Remember to keep track of signs based on the quadrant.

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 secant, allows calculation of others using identities like sec θ = 1/cos θ and tan²θ + 1 = sec²θ.

Sign of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant of the angle θ. In quadrant II, sine is positive, cosine and secant are negative, and tangent and cotangent are negative. This knowledge helps determine the correct sign of each function value when calculating from given information.

Rationalizing Denominators

Rationalizing denominators involves eliminating radicals from the denominator of a fraction by multiplying numerator and denominator by a suitable expression. This process simplifies the expression and is often required for final answers in trigonometry to maintain standard form.