Use a calculator to find each value. Give answers as real numbers.
cos (tan⁻¹ 0.5)

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Recognize that the expression involves a composition of functions: \(\cos(\tan^{-1}(0.5))\). Here, \(\tan^{-1}(0.5)\) is the angle whose tangent is 0.5.

Let \(\theta = \tan^{-1}(0.5)\). This means \(\tan(\theta) = 0.5\). We want to find \(\cos(\theta)\).

Recall the identity relating tangent and cosine: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). Since \(\tan(\theta) = 0.5\), we can think of a right triangle where the opposite side is 0.5 and the adjacent side is 1.

Use the Pythagorean theorem to find the hypotenuse: \(\text{hypotenuse} = \sqrt{(0.5)^2 + 1^2} = \sqrt{0.25 + 1} = \sqrt{1.25}\). Then, \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{1.25}}\).

Finally, use a calculator to evaluate \(\frac{1}{\sqrt{1.25}}\) to get the numerical value of \(\cos(\tan^{-1}(0.5))\).

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Key Concepts

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

Inverse Tangent Function (tan⁻¹ or arctan)

The inverse tangent function returns the angle whose tangent is a given number. For example, tan⁻¹(0.5) gives the angle θ such that tan(θ) = 0.5. This angle is typically measured in radians or degrees and lies within the principal range of -π/2 to π/2.

Relationship Between Trigonometric Functions

Understanding how sine, cosine, and tangent relate in a right triangle is essential. Given tan(θ) = opposite/adjacent, you can find cos(θ) by using the Pythagorean identity or by constructing a right triangle with sides representing the ratio, then calculating the adjacent side over the hypotenuse.

Using a Calculator for Trigonometric Values

Calculators can compute inverse trigonometric functions and standard trig functions directly. To find cos(tan⁻¹(0.5)), first calculate tan⁻¹(0.5) to get the angle, then use the cosine function on that angle. Ensure the calculator is set to the correct angle mode (degrees or radians) for accurate results.

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