Problem 24 - Essential Calculus

Question

Writing In Exercises 23 and 24 , give a short explanation of why the approximation is valid. $$ \tan 0.05 \approx 0+1(0.05) $$

Step-by-Step Solution

Answer

The approximation is valid due to the properties of small angle approximations. For small angles in radians, the tangent of the angle is approximately equal to the angle itself. This is the principle being applied to the given equation \( \tan 0.05 \approx 0.05 \).

Step 1: Understand small angle approximations

For relatively small angles, certain trigonometric functions can be well approximated by their respective angle measures in radians. This is particularly the case for the tangent function. The tangent of a very small angle is approximately equal to the angle itself when measured in radians. This is due to radians being the natural measure of angles which relates angle measure directly to arc length.

Step 2: Relate to the given equation

In the given equation, it's mentioned that \( \tan 0.05 \approx 0 + 1(0.05) \). Here, 0.05 is a small angle in radian measure. Applying the linear approximation for very small angles to this equation, one could say that \( \tan 0.05 \approx 0.05 \) which is exactly what the equation states.

Step 3: Final connection

Therefore, the given equation is a demonstration of the small angle approximation for the tangent function. The '1' in the equation is the coefficient of the angle measure, affirming that the tangent of a small angle in radian measure is approximately equal to the angle measure itself. The '0' in the equation serves as the constant term meaning the tangent of 0 is 0 which aligns to the fact that at 0 radian angle, the tangent is indeed 0.

Key Concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. It first arises from the ratio in a right triangle of the length of the opposite side to the adjacent side. In trigonometry, it has significant implications, especially with small angle approximations.

When dealing with small angles in radians, the tangent function simplifies considerably. This is because, for angles close to zero, the value of \( \tan(\theta) \) closely matches the angle \( \theta \) itself. This approximation makes calculations easier and more intuitive, relying on the linear behavior of the tangent function at very small ranges.

For example, \( \tan(0.05) \) where \( 0.05 \) is in radians, can be approximated as \( \tan(0.05) \approx 0.05 \), which aligns with our scenario of small angle approximation. The tangent function grows sharply and is not linear over its entire range, but when confined to such small angles, it behaves almost like a linear function.

Radian Measure

Radian measure is a way of measuring angles that directly relates the angle to the geometry of a circle. Unlike degrees, which are based on dividing one full rotation into 360 parts, radians relate angle size directly to the arc length of a circle.

A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. There are \( 2\pi \) radians in a complete circle. This measure is considered more natural for mathematical calculations because it integrates seamlessly with other mathematical concepts like arc length.

Understanding radian measure is important for grasping small angle approximations, especially with trigonometric functions like tangent. When angles are expressed in radians, trigonometric calculations can take advantage of mathematical properties and theorems that do not hold as simply under degree measurement.

• 1 radian is approximately 57.3 degrees.
• Small angles in radians simplify many trigonometric expressions.

Trigonometric Functions

Trigonometric functions are fundamental in understanding the relationships between the angles and sides of triangles, particularly right triangles. These include sine, cosine, tangent, and their reciprocal functions cosecant, secant, and cotangent.

Each trigonometric function has a specific role. For example, the sine function relates the ratio of the opposite side to the hypotenuse, while the tangent function involves the opposite side to the adjacent side.

In the realm of small angles, trigonometric functions can be approximated for simpler calculations. For very small angles, sine \( (\sin \theta \approx \theta) \), cosine \( (\cos \theta \approx 1) \), and tangent \( (\tan \theta \approx \theta) \) can be approximated when \( \theta \) is expressed in radians. This relationship emerges from the Taylor series expansion and simplifies calculations.

• With increasing angle size, the behavior of these functions deviates from linear approximations.
• Knowing these functions and their approximations benefits various mathematical and physics applications.