Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7.
sin θ = √2/6 , and cos θ < 0

Verified step by step guidance

1. Identify the given information: \(\sin \theta = \frac{\sqrt{2}}{6}\) and \(\cos \theta < 0\). This tells us the sine value and that the cosine is negative, which helps determine the quadrant of \(\theta\).
2. Recall the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Use this to find \(\cos \theta\) by substituting the given sine value: \(\left(\frac{\sqrt{2}}{6}\right)^2 + \cos^2 \theta = 1\).
3. Calculate \(\cos^2 \theta\) from the equation and then take the square root to find \(\cos \theta\). Since \(\cos \theta < 0\), choose the negative root.
4. Find \(\tan \theta\) using the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values found for sine and cosine.
5. Calculate the reciprocal functions: \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Rationalize denominators where necessary.

Key Concepts

Definition of the Six Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios of sides in a right triangle or coordinates on the unit circle. Given sin θ, the other functions can be found using their relationships, such as tan θ = sin θ / cos θ and reciprocal identities like csc θ = 1 / sin θ.

Using the Pythagorean Identity to Find Missing Values

The Pythagorean identity, sin²θ + cos²θ = 1, allows calculation of an unknown trigonometric value when one is given. Here, knowing sin θ and the sign of cos θ helps determine cos θ accurately, which is essential for finding all six function values.

Sign Determination Based on Quadrants

The sign of trigonometric functions depends on the quadrant where angle θ lies. Since sin θ is positive and cos θ is negative, θ is in the second quadrant. This information guides the correct sign assignment for all six functions, ensuring accurate results.