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

검증된 단계별 안내

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 \(\theta\).

Recall the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Use this to find \(\cos \theta\) by substituting the given sine value: \(\left(\frac{\sqrt{2}}{6}\right)^2 + \cos^2 \theta = 1\).

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.

Find \(\tan \theta\) using the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values found for sine and cosine.

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.

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주요 개념

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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.

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