Find the remaining five trigonometric functions of θ.
cos θ = 1/5, θ in quadrant I

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Identify the given information: \(\cos \theta = \frac{1}{5}\) and \(\theta\) is in Quadrant I, where all trigonometric functions are positive.

Use the Pythagorean identity to find \(\sin \theta\). Recall that \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\cos \theta = \frac{1}{5}\) into the identity: \(\sin^2 \theta + \left(\frac{1}{5}\right)^2 = 1\).

Solve for \(\sin \theta\): \(\sin^2 \theta = 1 - \left(\frac{1}{5}\right)^2 = 1 - \frac{1}{25}\). Then, \(\sin \theta = \sqrt{1 - \frac{1}{25}}\). Since \(\theta\) is in Quadrant I, take the positive root.

Find \(\tan \theta\) using the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) found in previous steps.

Calculate the reciprocal functions: \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Use the values obtained for \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1. Given cos θ, this identity allows us to find sin θ by rearranging the equation to sin θ = √(1 - cos²θ). This is essential for determining the sine value when cosine is known.

Signs of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant of the angle. Since θ is in quadrant I, all six trigonometric functions (sin, cos, tan, cot, sec, csc) are positive. This information helps assign the correct sign to the calculated values.

Definitions of Trigonometric Functions

The six trigonometric functions are defined as ratios of sides in a right triangle or as reciprocal functions: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ / cos θ, cot θ = 1 / tan θ, sec θ = 1 / cos θ, and csc θ = 1 / sin θ. Knowing these definitions allows calculation of all functions once sin θ and cos θ are known.

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