Question

Which of the following trigonometric functions is equivalent to the function \(g(x)=\sin x \sec x ?\) \(\left.\text { (Note: } \sec x=\frac{1}{\cos x}\right)\) A. \(f(x)=\cos x\) B. \(f(x)=\cot x\) C. \(f(x)=\csc x\) D. \(f(x)=\sin x\) E. \(f(x)=\tan x\)

Short answer

Answer: (E) \(f(x) = \tan x\)

Step-by-step solution

Simplify the given function

Since \(\sec x = \frac{1}{\cos x}\), the given function can be rewritten as: \(g(x) = \sin x \cdot \frac{1}{\cos x}\).

Express the function in terms of ratio

Now, we can simplify the expression by multiplying \(\sin x\) with \(\frac{1}{\cos x}\):\( g(x) = \frac{\sin x}{\cos x}\)

Identify the equivalent option

To find which option is equivalent to the simplified function, \(g(x) = \frac{\sin x}{\cos x}\), we check each option: A. \(f(x)=\cos x\) => Not equivalent B. \(f(x)=\cot x\) => \(\cot x = \frac{\cos x}{\sin x}\) => Not equivalent C. \(f(x)=\csc x\) => \(\csc x = \frac{1}{\sin x}\) => Not equivalent D. \(f(x)=\sin x\) => Not equivalent E. \(f(x)=\tan x\) => \(\tan x = \frac{\sin x}{\cos x}\) => Equivalent So, the option (E) \(f(x)=\tan x\) is equivalent to the function \(g(x)=\sin x \sec x\).

Key concepts

Sine and Cosine

The sine and cosine functions are fundamental to trigonometry, often depicted as the ratios of sides in a right-angled triangle. For an angle \( \theta \), the sine function, denoted \( \sin(\theta) \), represents the ratio of the length of the side opposite to the angle to the length of the hypotenuse. In contrast, the cosine function, represented as \( \cos(\theta) \), corresponds to the ratio of the length of the adjacent side to the hypotenuse.

These functions are also defined in the unit circle context, where each point on the circumference is the representation of an angle's sine and cosine values on the x and y-axes, respectively. This perspective allows us to extend the domain of these functions beyond the limitations of a triangle to all real numbers, encapsulating full rotational measurement.

In practical application, understanding the relationship between sine and cosine is crucial. They are phase-shifted versions of each other, and in many cases, you can express one function in terms of the other, such as \( \sin(\theta) = \cos(90^\circ - \theta) \) and vice versa.

Secant Function

The secant function, abbreviated as \( \sec \), might be less well-known but is just as significant in the family of trigonometric functions. It is the reciprocal of the cosine function, meaning that for any angle \( \theta \), it is defined as \( \sec(\theta) = \frac{1}{\cos(\theta)} \), provided that \( \cos(\theta) \) is not zero to avoid undefined values.

Graphically, the secant function exhibits a wave-like structure with vertical asymptotes where the cosine function is zero. This reflects the concept that as the cosine approaches zero, the secant function's value becomes infinitely large. In the context of a right-angled triangle, if one were to imagine extending the adjacent side outside of the triangle to meet a line parallel to the hypotenuse, the secant represents the ratio of the length of this extended line to the hypotenuse.

The significance of the secant function escalates in advanced mathematics, where it's used in calculus for various integration purposes and to solve specific equations that are not easily approachable using only sine or cosine.

Tangent Function

The tangent function, or \( \tan \), is another primary trigonometric function that combines the concepts of sine and cosine. It is defined as the ratio of the sine function to the cosine function, mathematically represented as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

Understanding Tangent in Triangles

In the context of a triangle, \( \tan(\theta) \) can be interpreted as the ratio of the opposite side to the adjacent side for a given angle \( \theta \) within a right-angled triangle. This ratio is particularly useful because it's independent of the triangle's size, relying solely on the angle, providing a universal relationship that applies to right-angled triangles.

Application in Real-Life Problems

Tangent is employed in several real-world applications, such as determining the height of an inaccessible object by measuring an angle and a distance, which is especially useful in fields like surveying and navigation. Similarly, in calculus, tangent lines to curves play a pivotal role in analyzing and understanding the properties of functions. By comprehending the tangent's behavior, especially in relation to sine and cosine, students can tackle a wide range of trigonometric problems effectively.