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Chapter 1: Problem 47

Use a calculator to find the radian measure of an acute angle whose trigonometric function is given. \(\cos t=0.60\)

Short Answer

Expert verified
t 鈮� 0.927 radians

Step by step solution

01

Understand the Problem

The goal is to find the radian measure of an acute angle, denoted as t, for which the cosine is 0.60. This means solving for t in the equation \( \cos t = 0.60 \).
02

Use the Inverse Cosine Function

To find the angle t in radians whose cosine is 0.60, use the inverse cosine function. This can be written as \( t = \arccos(0.60) \).
03

Calculate with a Calculator

Enter \( \arccos(0.60) \) into a calculator to find the value. Make sure the calculator is set to radians. The value should be approximately 0.927 radians.

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Key Concepts

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

Understanding Acute Angles
An acute angle is one of the basic concepts in geometry and trigonometry. It is any angle that measures less than 90 degrees. This means it's smaller than a right angle.
In the context of trigonometric functions, such as cosine, sine, and tangent, acute angles often arise. Here are some key points to remember about acute angles:
  • They are always positive and less than 90 degrees or \( \frac{\pi}{2} \) radians.
  • They appear frequently in triangles, especially in right-angled triangles where the two non-right angles are acute.
  • In the unit circle, which is a crucial tool in trigonometry, acute angles are found in the first quadrant.
Understanding acute angles helps in solving trigonometric problems, like the one in our exercise. We need the angle whose cosine is 0.60, and it must be acute.
Exploring the Cosine Function
The cosine function is a fundamental trigonometric function. It's denoted as \( \cos \) and is one of the basic building blocks for understanding angles and their relationships.

The cosine of an angle gives the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. Let's break down some essential points:
  • The cosine function is an even function, meaning \( \cos(-x) = \cos(x) \).
  • Its values range from -1 to 1.
  • For acute angles, the cosine values are positive and lie between 0 and 1.
In our exercise, we use the inverse cosine (or arccos) to find the angle. Given that \( \cos t = 0.60 \), we write it as \( t = \arccos(0.60) \). Use a calculator set to radians to find that \( t \) is approximately 0.927 radians.
Radian Measure
Radian measure is a way of measuring angles used mainly in trigonometry. It's different from degrees but provides a natural way to describe angles in terms of pi (\( \pi \)).
One radian is the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius. Here's why radians are important:
  • They simplify many trigonometric formulas, making them elegant and easy to remember.
  • There are \( 2\pi \) radians in a full circle (360 degrees).
  • The connection between radian measures and functions like \( \sin \) and \( \cos \) is more straightforward in calculus and advanced math.
In the exercise, we found \( t \) using a calculator and the \( \arccos \) function to be approximately 0.927 radians, which is an acute angle measurement. Radians help in naturally understanding and computing trigonometric values.

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Most popular questions from this chapter

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Use a calculator to find the degree measure of an acute angle whose trigonometric function is given. \(\sin t=0.45\)
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