Textbook Question
Evaluate each expression without using a calculator.
sec (sec⁻¹ 2)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 91
Textbook Question
Use a calculator to find each value. Give answers as real numbers.
cos (tan⁻¹ 0.5)
Verified step by step guidance1
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 \(\cos(\tan^{-1}(0.5))\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
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.
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Inverse Tangent
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.
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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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