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
Recognize that the expression is \( \csc(\tan^{-1}(\frac{\sqrt{3}}{3})) \). The first step is to let \( \theta = \tan^{-1}(\frac{\sqrt{3}}{3}) \), which means \( \tan(\theta) = \frac{\sqrt{3}}{3} \).
Recall that \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Assign the opposite side as \( \sqrt{3} \) and the adjacent side as 3 to form a right triangle representing \( \theta \).
Use the Pythagorean theorem to find the hypotenuse \( h \) of the triangle: \( h = \sqrt{(\sqrt{3})^2 + 3^2} = \sqrt{3 + 9} = \sqrt{12} \). Simplify \( \sqrt{12} \) to \( 2\sqrt{3} \).
Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \) and \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), calculate \( \sin(\theta) = \frac{\sqrt{3}}{2\sqrt{3}} \).
Finally, express \( \csc(\theta) \) as the reciprocal of \( \sin(\theta) \), which is \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Simplify this expression to find the exact value.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, like tan⁻¹(x), return the angle whose trigonometric ratio equals x. For example, tan⁻¹(√3/3) gives the angle whose tangent is √3/3, which helps in finding the angle measure needed to evaluate other trigonometric functions.
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Right Triangle Trigonometry
Right triangle trigonometry relates the sides of a right triangle to its angles using ratios like sine, cosine, and tangent. Knowing the tangent ratio allows you to determine the opposite and adjacent sides, which can then be used to find other ratios such as cosecant.
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Reciprocal Trigonometric Functions
Cosecant (csc) is the reciprocal of sine, defined as csc(θ) = 1/sin(θ). Once the angle θ is found, calculating csc(θ) involves finding the sine of θ and then taking its reciprocal to get the exact value.
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