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Chapter 1: Problem 31

For an acute angle \(A\), can \(\cos A\) be larger than 1 ? Explain your answer.

Short Answer

Expert verified
No, \(\text{cos}(A)\) cannot be larger than 1 for any acute angle because the range of cosine for acute angles is between 0 and 1.

Step by step solution

01

Understand the Range of Cosine Function

The cosine function, \(\text{cos}(A)\), is defined for all real numbers and oscillates between -1 and 1 for all angles. Specifically, \(\text{cos}(A)\) is always between -1 and 1 inclusive.
02

Define the Acute Angle

An acute angle is any angle \(A\) that is between 0 and 90 degrees (exclusive). This means that the angle is strictly positive and less than 90 degrees.
03

Evaluate Cosine in Acute Angles

For any acute angle \(A\), \(\text{cos}(A)\) will always be positive because cosine values for angles between 0 and 90 degrees are positive and range from 1 to 0 as the angle increases.
04

Compare \(\text{cos}(A)\) to 1

Since \(\text{cos}(A)\) for acute angles ranges from 1 to 0, it cannot exceed 1. Therefore, it is not possible for \(\text{cos}(A)\) to be larger than 1.
05

Conclusion

Given the properties of the cosine function and the definition of acute angles, \(\text{cos}(A)\) cannot be larger than 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Range
The cosine function, represented as \(\text{cos}(A)\), plays a crucial role in trigonometry. This function measures the horizontal component of an angle in a right triangle. Cosine values oscillate between -1 and 1 for all possible angles, regardless of the quadrant in which the angle lies. Let's break this down:
  • For any angle, \(\text{cos}(A)\) cannot be less than -1 or greater than 1.
  • This characteristic holds true for both positive and negative angles.
  • The highest value \(\text{cos}(A)\) can attain is 1, which occurs when the angle is exactly 0 degrees or any multiple of 360 degrees.
Because of this inherent property, \(\text{cos}(A)\) is bounded and will never exceed these limits.
Acute Angle Definition
An acute angle is defined as an angle that is greater than 0 degrees but less than 90 degrees. These angles are found in various real-life applications, such as the angles of a cutting blade or the incline of a hill. Here are a few key points about acute angles:
  • Acute angles are always positive.
  • They range from just above 0 degrees up to 89.999... degrees.
  • They are most commonly found in right triangles, where one angle is always 90 degrees, and the remaining two must sum to 90 degrees, making them both acute.
When we consider the cosine of an acute angle, we are always working within this positive angle range, which directly influences the value of \(\text{cos}(A)\).
Trigonometric Function Properties
Trigonometric functions such as sine, cosine, and tangent have specific properties that are essential for solving many math problems. Focusing on the cosine properties now:
  • The cosine function is even, meaning \(\text{cos}(-A) = \text{cos}(A)\).
  • Cosine values are positive in the first and fourth quadrants and negative in the second and third quadrants.
  • The value of \(\text{cos}(A)\) for angles between 0 and 90 degrees ranges from 1 to 0. This range does not include values greater than 1 or less than 0.
Because acute angles fall strictly within the 0 to 90 degrees range, the cosine function remains positive and, importantly, within the defined range of 1 to 0. Therefore, for any acute angle \(\text{cos}(A)\), the maximum value will be 1, and it will decrease towards 0 as the angle approaches closer to 90 degrees.

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Most popular questions from this chapter

In which quadrant(s) do sine and cosine have the opposite sign?

State in which quadrant or on which axis the given angle lies. $$-127^{\circ}$$

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