Problem 27 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

. \(\cos 14^{\circ}\)

Step-by-Step Solution

Answer

To find the precise value of \(\cos 14^\circ\), use a scientific calculator. The answer depends on the calculator's precision.

Step 1: Identify the Function and the Angle

The function we are working with is the cosine function, denoted by \(\cos\). The angle in question is \(14^\circ\).

Step 2: Use a Scientific Calculator

The precise value for cosine of angles not typically found on trigonometric tables, like \(14^\circ\), is not commonly memorized. Therefore, you need to use a scientific calculator. Input \(14\) into the calculator and then press the \(\cos\) button.

Step 3: Interpret the Result

The calculator will provide the result. This is the value of \(\cos 14^\circ\).

Key Concepts

Trigonometric Functions

Understanding trigonometric functions is essential when studying the relationships within a right triangle or the properties of waves and oscillations. The cosine function, one of the primary trigonometric functions, relates a given angle to the ratio of the adjacent side over the hypotenuse in a right-angled triangle.

It is represented as \( \cos(\theta) \) where \( \theta \) is the angle in question. Besides cosine, there are sine and tangent functions, collectively known to be the three main trigonometric functions, which each give different ratios of a triangle's sides. For example, sine is the ratio of the opposite side to the hypotenuse, while tangent is the ratio of the opposite side to the adjacent side.

These functions are fundamental in various areas of math and science, such as physics, engineering, and even computer graphics, where they help describe the shapes and motions of objects.

Scientific Calculator Usage

A scientific calculator is a valuable tool for solving mathematical problems that go beyond simple arithmetic. When dealing with trigonometric functions like the cosine, a scientific calculator helps you find precise values quickly. To use it for our exercise, input the angle value, in this case \(14^\circ\) and then press the corresponding function, \(\cos\), to get the result.

It's important to ensure that the calculator is set to the correct mode (degrees or radians) corresponding to the measure of the angle you're working with. For angles measured in degrees, make sure your calculator is set to degree mode, not radian mode, otherwise, you'll get an incorrect result.

Common features of scientific calculators:

• Capability to perform exponentiation and root extractions.
• Functions for trigonometry, logarithms, and probability.
• Complex number computations

If you're ever uncertain about how to use a specific function on your calculator, consult the manual for detailed instructions.

Trigonometry

Trigonometry is a branch of mathematics that deals with the study of triangles, particularly right-angled triangles, and the relationships between their angles and sides. It is a subject that blends geometry and algebra, allowing us to solve for the lengths of sides or the measures of angles within a triangle.

Applications of trigonometry are diverse, ranging from calculating distances in navigation and astronomy, analyzing sound waves, designing mechanical structures, and even in the development of computer algorithms.

Basic Trigonometric Identities:

• Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Angle sum and difference identities.
• Double angle formulas.

By understanding the principles of trigonometry, one can approach complex problems with a toolkit of methods to analyze and solve them, whether that be in the math classroom or in professional technical fields.