Question

Given \(\csc ^{2} x+\sin ^{2} x=7.89,\) find the value of \(\frac{1}{\csc ^{2} x}+\frac{1}{\sin ^{2} x}\).

Short answer

The value is 7.89.

Step-by-step solution

- Recap Trigonometric Identities

Recall the identity for cosecant: \ \( \csc x = \frac{1}{sin x} \)

- Rewrite Given Equation

Rewrite the given equation in terms of sine and its reciprocal. The given equation is: \ \( \csc^2 x + sin^2 x = 7.89 \).\ \csc^2 x can be expanded as \frac{1}{sin^2 x}, so the equation becomes: \ \( \frac{1}{sin^2 x} + sin^2 x = 7.89 \)

- Assign Temporary Variables

Let \sin^2 x = y \ and \csc^2 x = z \. So \ \( y + z = 7.89 \) where \ \( z = \frac{1}{y} \). Substitute \ \( z = \frac{1}{y} \) into the equation: \ \( y + \frac{1}{y} = 7.89 \)

- Find the Required Expression

We need to find \frac{1}{\frac{1}{y}} + \frac{1}{y} \ which simplifies to: \ \( y + z \). Since we already have that \ \( y + z = 7.89 \), the required value is 7.89

Key concepts

Cosecant Function

The cosecant function, denoted as \(\text{csc } x\), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. This means it represents the ratio of the hypotenuse to the opposite side in a right triangle for a given angle, x. Mathematically, this relationship is written as:
\[ \text{csc } x = \frac{1}{\text{sin } x} \]
In the context of trigonometry, understanding the cosecant function is crucial because it helps to solve various equations involving angles and lengths of triangles. Remember that because cosecant is a reciprocal of sine, wherever sine is zero, cosecant is undefined.

Reciprocal Identities

Reciprocal identities are an essential part of trigonometry that express relationships between trigonometric functions and their reciprocals. For example, as mentioned earlier, the reciprocal of the sine function \( \text{sin } x \) is the cosecant function \( \text{csc } x \). Similarly, other trigonometric functions have their reciprocals, such as:

• \[ \text{sec } x = \frac{1}{\text{cos } x} \]
• \[ \text{cot } x = \frac{1}{\text{tan } x} \]

These identities are crucial for transforming and solving trigonometric equations. They enable us to rewrite expressions in more manageable forms, often simplifying the problem at hand. Understanding and utilizing these identities can help you solve complex trigonometric problems more effectively. For example, in our exercise, we used the reciprocal identity \( \text{csc } x = \frac{1}{\text{sin } x} \) to rewrite and solve the equation.

Sine Function

The sine function, denoted as \( \text{sin } x \), is one of the primary trigonometric functions. It represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. Mathematically, it is expressed as:
\[ \text{sin } x = \frac{\text{opposite}}{\text{hypotenuse}} \]
The sine function is periodic, with a period of \( 2\text{π} \), indicating that its values repeat every \( 2\text{π} \) units. Its range lies between -1 and 1, inclusive. The sine function is fundamental in various fields, including physics, engineering, and computer science.
When solving trigonometric equations, knowing the properties and behavior of the sine function can simplify your work. In our given exercise, recognizing that \( \text{csc } x \) is the reciprocal of \( \text{sin } x \) was essential to derive the solution.