Problem 42 - College Trigonometry

Question

For the given vector \(\vec{v}\), find the magnitude \(\|\vec{v}\|\) and an angle \(\theta\) with \(0 \leq \theta<360^{\circ}\) so that \(\vec{v}=\|\vec{v}\|\langle\cos (\theta), \sin (\theta)\rangle\) (See Definition 11.8.) Round approximations to two decimal places. $$ \vec{v}=\langle 3,4\rangle $$

Step-by-Step Solution

Answer

Magnitude is 5, angle is approximately 53.13°.

Step 1: Calculate the Magnitude

The magnitude of the vector \( \vec{v} = \langle 3, 4 \rangle \) is given by the formula \( \|\vec{v}\| = \sqrt{x^2 + y^2} \). Here, \( x = 3 \) and \( y = 4 \). Thus, \( \|\vec{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).

Step 2: Calculate the Angle

The vector \( \vec{v} \) can be represented as \( \langle \cos(\theta), \sin(\theta) \rangle \). The angle \( \theta \) can be found using the tangent function where \( \tan(\theta) = \frac{y}{x} = \frac{4}{3} \). Therefore, \( \theta = \tan^{-1}\left(\frac{4}{3}\right) \). Using a calculator, \( \theta \approx 53.13^{\circ} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles of a right triangle to the lengths of its sides. They are vital in understanding vector components and calculating angles in vector problems. The primary trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)).
\(\sin(\theta)\) is the ratio of the length of the opposite side to the hypotenuse, \(\cos(\theta)\) is the ratio of the adjacent side to the hypotenuse, and \(\tan(\theta)\) is the ratio of the opposite side to the adjacent side.
For vectors, we often express a vector in terms of trigonometric functions to find either the angle or other missing vector components. This approach is particularly useful when a vector is broken into its horizontal and vertical components, which align with \(\cos(\theta)\) and \(\sin(\theta)\) respectively.

Vector Components

Vectors have both magnitude and direction, which can be represented in terms of components. In a 2D plane, any vector can be broken down into two perpendicular components: horizontal and vertical.
This is often represented as \(\langle x, y \rangle\), where \(x\) is the horizontal component and \(y\) is the vertical component. These components can be found using trigonometry.
The formula \(\vec{v} = \langle \cos(\theta), \sin(\theta) \rangle\) links the directional movement of a vector with trigonometric functions, associating \(\cos(\theta)\) with the horizontal distance and \(\sin(\theta)\) with the vertical distance.

• The magnitude of the vector components can be calculated using the Pythagorean theorem: \(\|\vec{v}\| = \sqrt{x^2 + y^2}\).
• The angle \(\theta\) is the angle between the vector and the positive x-axis, and understanding the relation to its components helps in tasks such as navigation and physics problems.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when the sides of a right triangle are known. These functions are essential when solving for unknown angles in vectors.
The primary inverse trigonometric functions include arcsine (\(\sin^{-1}\)), arccosine (\(\cos^{-1}\)), and arctangent (\(\tan^{-1}\)).
For example, to find the angle \(\theta\) in our vector problem, we use the arctangent function. Since \(\tan(\theta) = \frac{y}{x}\), \(\theta\) can be determined by \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).
This inverse function yields the angle whose tangent is a given value, making it crucial for finding angular directions in 2D vector components. When using calculators, ensure the correct mode (degrees or radians) to get accurate calculations.