Question

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{\theta \rightarrow 0} \frac{\cos \theta \tan \theta}{\theta} $$

Short answer

The limit of the function \(\frac{\cos \theta \tan \theta}{\theta}\) as \( \theta \rightarrow 0\) is \(1\).

Step-by-step solution

Use a trigonometric identity

We start by rewriting the function using the identity that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), so the function \(\frac{\cos \theta \tan \theta}{\theta}\) becomes \(\frac{\cos \theta \cdot (\frac{\sin \theta}{\cos \theta})}{\theta} = \frac{\sin \theta}{\theta}\).

Apply the limit theorem

We know that \(\lim_{\theta \rightarrow 0} \frac{\sin \theta}{\theta} = 1\), a famous limit used often in calculus. So, apply this to our given function, and regard \(\theta\) approaches towards 0, the function \(\frac{\sin \theta}{\theta}\) will equals to 1.

Conclusion

Now, adding the results from step 1 and step 2, we can conclude that the limit of the original function \(\lim _{\theta \rightarrow 0} \frac{\cos \theta \tan \theta}{\theta}\) is \(1\).

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially within the field of calculus. These functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)), which are essential in defining the relationship between the angles and sides of a right triangle. They portray periodic behavior, which is crucial in understanding how angles and their corresponding values interact over intervals.

• Sine represents the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.
• Cosine indicates the ratio of the length of the adjacent side to the hypotenuse.
• Tangent combines these functions, being the ratio of sine to cosine.

Understanding these functions' properties is essential, as it allows you to manipulate and solve complex problems involving waves, rotations, and other periodic phenomena. Their continuous and smooth nature makes them suitable for applying limits as angles approach specific values, as seen in calculus exercises.

Limit Theorems

Limit theorems form the backbone of calculus. They provide a way to understand how a function behaves as its input approaches a particular value. When it comes to trigonometric functions, these theorems allow us to explore their behavior as angles tend to zero or infinity, which often shows up in real-world applications.- The **Limit Theorem** for the sine function \(\lim_{\theta \rightarrow 0} \frac{\sin \theta}{\theta} = 1\) is a well-known result used in various calculus problems. This theorem depicts how the ratio of the sine of an angle to the angle itself approaches 1 when the angle is in radians. - Similar limits apply to other trigonometric functions, offering insights into how they behave for very small or very large inputs.Applying these limit theorems allows us to simplify expressions and solve problems that would otherwise be too complex or impossible to evaluate directly. They are fundamental tools in understanding how functions behave near their boundary or critical points.

Calculus

Calculus is the branch of mathematics concerned with change and motion. It's composed of two main ideas: differentiation, which deals with the rate of change, and integration, which deals with the accumulation of quantities. Calculus is widely used to solve problems in physics, engineering, economics, and beyond.One of the critical applications of calculus is evaluating the limits of functions.

• In the problem where we find the limit of the trigonometric function \( \lim_{\theta \rightarrow 0} \frac{\cos \theta \tan \theta}{\theta} \), we use calculus to simplify and solve it.
• Differentiation applies here by recognizing the rate at which the function approaches a limit as the input \(\theta\) approaches zero.
• Integration may not seem directly related in this context but is a powerful tool for handling problems of accumulation and area under curves.

By understanding these processes, you gain the ability to analyze how functions behave and change, providing valuable insights into practical and theoretical problems in mathematics and beyond. Calculus is thus an indispensable part of modern science and engineering.