The Point P (4, 2) on the Terminal Side of Angle θ in Standard Position

The given problem involves determining the cosine value of an angle θ that is in standard position, given that the point P (4, 2) is on the terminal side of this angle. This example utilizes basic trigonometric principles to find the required cosine value. This article will guide us through the process of solving such problems and explain the key terms and concepts involved.

Understanding Angles in Standard Position

An angle is said to be in standard position when its vertex is at the origin (0, 0) of the coordinate system, and its initial side lies along the positive x-axis. The terminal side is the side that lies in the direction determined by the rotation of the angle.

The position of any point P on the terminal side can be represented by its coordinates (x, y). For the given point P (4, 2), we need to determine the values of x, y, and r, where r is the distance from the origin to point P, also known as the radius or hypotenuse in a right triangle.

Calculating the Distance r

The distance r can be calculated using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (r) is equal to the sum of the squares of the other two sides (x and y). The formula is given as:

[ r sqrt{x^2 y^2} ]

For the given point P (4, 2), substituting the x and y values into the formula, we get:

[ r sqrt{4^2 2^2} ]

Which simplifies to:

[ r sqrt{16 4} sqrt{20} 2sqrt{5} ]

Calculating the Cosine of Angle θ

The cosine of an angle θ in standard position in a right triangle can be defined as the ratio of the length of the adjacent side (x) to the hypotenuse (r). The formula for cosine is:

[ cos{theta} frac{x}{r} ]

Substituting the values of x and r, we get:

[ cos{theta} frac{4}{2sqrt{5}} ]

This simplifies to:

[ cos{theta} frac{2}{sqrt{5}} ]

However, it is often desirable to rationalize the denominator for clarity and simplicity. Therefore, we multiply the numerator and the denominator by √5:

[ cos{theta} frac{2sqrt{5}}{5} ]

Summary

In conclusion, given that the point P (4, 2) is on the terminal side of an angle θ in standard position, the cosine of θ is [ cos{theta} -frac{2sqrt{5}}{5} ]. This solution is derived from basic trigonometry principles and demonstrates the use of the Pythagorean theorem and the definition of cosine to solve for unknown values in right triangles.

Key Concepts and Terms

Angle in standard position: An angle with its vertex at the origin and its initial side along the positive x-axis. Terminal side: The rotation direction determined by the angle. Cosine: The ratio of the adjacent side to the hypotenuse in a right triangle. Pythagorean theorem: A^2 B^2 C^2 in a right triangle, where C is the hypotenuse. Rationalizing the denominator: The process of removing radicals from the denominator of a fraction.

Benefits of Understanding Trigonometry in Digital Marketing

In the context of SEO and digital marketing, understanding trigonometry can be beneficial for optimizing website design and improving user experience. For example:

Optimizing imagery: Trigonometry can help in determining the correct proportions of images, ensuring they are displayed correctly across different devices and orientations. Improving website layout: Knowledge of angles and the cosine function can aid in designing responsive web pages that adapt to various screen sizes. Enhancing user experience: Understanding angles and their relationships can improve visual elements, creating a more aesthetically pleasing and engaging experience for users.

By mastering these concepts and their applications, marketers can create more effective and engaging digital content that resonates with their target audience.