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Chapter 1: Problem 48

\(\cos (-2.5)\)

Short Answer

Expert verified
The exact value of \(\cos (-2.5)\) is approximately 0.8011436155469337. For practical purposes, you may round it off depending on the number of significant figures required. For instance, if three significant figures are needed, then the answer would be 0.801.

Step by step solution

01

Understand the Cosine function

Cosine is a trigonometric function which is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. But for this problem, there's no right triangle and the input isn't from a usual angle measure. We just need to know that cosine is a special function that cyclically oscillates as you go farther along the number line, and it can handle any sort of input, even a negative one like -2.5.
02

Use calculator or mathematical software

In the next step, you must use a calculator or mathematical software tool that can compute trigonometric functions. Make sure your calculator is set in radian mode (since the standard value for trigonometric function in mathematics is radian) and type \(\cos(-2.5)\) and press the 'equals' sign or 'enter'.
03

Record the Result

Finally, record the result provided by the calculator.

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Key Concepts

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

Trigonometric Functions
Trigonometric functions are fundamental in mathematics and are used to describe relationships in triangles, oscillations, and waves. The main trigonometric functions are sine, cosine, and tangent.
  • These functions help in understanding angles and dimensions in geometric shapes.
  • They play a crucial role in modeling periodic phenomena like sound and light waves.
The cosine function, in particular, is defined for any real number input. Unlike in traditional triangle definitions, cosine values are calculated using real numbers representing angles, even if you can't draw a triangle for them.

Additionally, cosine is known for its repetitive pattern: it completes one cycle every full circle, which is around 6.28 radians or 360 degrees. This is why trigonometric functions are excellent for describing cycles and waves.
Radians
Radians are a unit of angular measure used in many areas of mathematics. One radian is the angle created when the arc length is equal to the radius of the circle.
  • Radians offer a way to describe angles based on the radius of a circle, providing a natural and efficient system for measuring angles.
  • They are more commonly used in higher-level mathematics and certain fields like physics due to their simplicity in calculations.
The standard unit for measuring angles in trigonometry is radians rather than degrees. This ensures calculations are consistent and directly related to the properties of circles. In trigonometric functions, such as cosine, arguments are expressed in radians, making operations seamless and universal across mathematical contexts.
Mathematical Software
Calculating trigonometric functions manually can be intensive, especially when dealing with non-standard angles like 2.5 radians. This is where mathematical software and calculators come into play. They efficiently handle complex computations and ensure accuracy.
  • Use tools like graphing calculators, computer software like MATLAB, or online calculators to compute trigonometric values.
  • These tools typically allow users to toggle between degrees and radians, ensuring correct calculations.
  • Mathematical software often includes additional features to plot functions, analyze derivatives, and integrate more complex equations.
This means entering an expression like \( \cos(-2.5) \) becomes simple and quick. With a calculator in radian mode, you can effortlessly find the value, which shows the practicality and necessity of mathematical tools in contemporary problem-solving.

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Most popular questions from this chapter

Prove each identity. (a) \(\arcsin (-x)=-\arcsin x\) (b) \(\arctan (-x)=-\arctan x\) (c) \(\arctan x+\arctan \frac{1}{x}=\frac{\pi}{2}, \quad x>0\) (d) \(\arcsin x+\arccos x=\frac{\pi}{2}\) (e) \(\arcsin x=\arctan \frac{x}{\sqrt{1-x^{2}}}\)

In Exercises 109-112, sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side. Then find the other five trigonometric functions of \(\boldsymbol{\theta}\). $$ \sin \theta=\frac{3}{4} $$

In Exercises 19-34, use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arcsin \frac{3}{4} $$

When graphing the sine and cosine functions, determining the amplitude is part of the analysis. Explain why this is not true for the other four trigonometric functions.

$$ \text { In Exercises 49-58, find the exact value of the expression. } $$ $$ \cos \left(\tan ^{-1} 2\right) $$
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