Textbook Question
In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places.tan⁻¹ (−20)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
3:51 minutesProblem 43
Textbook Question
In Exercises 29–51, find the exact value of each expression. Do not use a calculator._csc(tan⁻¹ √3/3)
Verified step by step guidance1
Step 1: Understand the problem. We need to find the exact value of \( \csc(\tan^{-1}(\frac{\sqrt{3}}{3})) \).
Step 2: Recognize that \( \tan^{-1}(\frac{\sqrt{3}}{3}) \) represents an angle \( \theta \) such that \( \tan(\theta) = \frac{\sqrt{3}}{3} \).
Step 3: Recall that \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). So, we can consider a right triangle where the opposite side is \( \sqrt{3} \) and the adjacent side is \( 3 \).
Step 4: Use the Pythagorean theorem to find the hypotenuse: \( \text{hypotenuse} = \sqrt{(\sqrt{3})^2 + 3^2} = \sqrt{3 + 9} = \sqrt{12} = 2\sqrt{3} \).
Step 5: Recall that \( \csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} \). Substitute the values: \( \csc(\theta) = \frac{2\sqrt{3}}{\sqrt{3}} = 2 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, such as tan⁻¹, are used to find the angle whose tangent is a given value. In this case, tan⁻¹(√3/3) corresponds to an angle where the opposite side is √3 and the adjacent side is 3, which can be simplified to find the angle in a right triangle.
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Cosecant Function
The cosecant function, denoted as csc, is the reciprocal of the sine function. It is defined as csc(θ) = 1/sin(θ). To find csc(tan⁻¹(√3/3)), one must first determine the sine of the angle obtained from the inverse tangent function.
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Right Triangle Relationships
Understanding the relationships in a right triangle is crucial for solving trigonometric problems. By using the Pythagorean theorem, one can find the lengths of the sides of the triangle based on the known ratios, which helps in calculating sine, cosine, and cosecant values for the angle derived from the inverse tangent.
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