Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 2.3.10

Chapter 3, Problem 2.3.10

CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.

Column I: 1.
cot⁻¹ 30
Column II:
A. 88.09084757°
B. 63.25631605°
C. 1.909152433°
D. 17.45760312°
E. 0.2867453858
F. 1.962610506
G. 14.47751219°
H. 1.015426612
I. 1.051462224
J. 0.9925461516

Verified step by step guidance

Step 1: Understand the inverse trigonometric function involved. Here, \( \cot^{-1} 30 \) means the angle whose cotangent is 30.

Step 2: Recall the relationship between cotangent and tangent: \( \cot \theta = \frac{1}{\tan \theta} \). So, \( \cot^{-1} 30 = \theta \) implies \( \tan \theta = \frac{1}{30} \).

Step 3: To find the angle \( \theta \), use the inverse tangent function: \( \theta = \tan^{-1} \left( \frac{1}{30} \right) \).

Step 4: Calculate \( \tan^{-1} \left( \frac{1}{30} \right) \) using a calculator or trigonometric tables to find the angle in degrees or radians, depending on the context.

Step 5: Match the calculated angle with the closest approximation given in Column II to complete the pairing.

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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 cot⁻¹ (arccotangent), are used to find the angle whose trigonometric ratio equals a given value. For example, cot⁻¹(30) gives the angle whose cotangent is 30. Understanding these functions is essential for converting between angle measures and ratio values.

Angle Measurement and Approximation

Angles can be measured in degrees or radians and often require approximation for practical use. Matching trigonometric values to their angle approximations involves understanding how to interpret and convert between exact values and decimal approximations, which is crucial for solving problems involving trigonometric functions.

Trigonometric Ratios and Their Properties

Trigonometric ratios like sine, cosine, tangent, and cotangent relate the angles of a right triangle to the ratios of its sides. Knowing the properties and ranges of these ratios helps in identifying correct angle-value pairs and understanding the behavior of these functions across different quadrants.

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