What is 1 / sin x = cosec x? | Simple Explanation for Class 10

S6-SA2-0408

What is the Identity 1 / sin x = cosec x?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The identity 1 / sin x = cosec x tells us that cosecant (cosec x) is the reciprocal of sine (sin x). This means if you know the value of sin x, you can find cosec x by simply dividing 1 by sin x, and vice-versa. It's a fundamental relationship in trigonometry.

Simple Example

Quick Example

Imagine you have 1 litre of chai, and you want to pour it into cups. If each cup holds sin x amount of chai, then 1 / sin x tells you how many cups you can fill. Cosec x is just another way of saying the same thing – it represents the number of cups. So, if sin x is 0.5 (meaning half a litre), then cosec x = 1 / 0.5 = 2 cups.

Worked Example

Step-by-Step

PROBLEM: If sin x = 3/5, what is the value of cosec x?
---STEP 1: Recall the identity: cosec x = 1 / sin x.
---STEP 2: We are given sin x = 3/5.
---STEP 3: Substitute the value of sin x into the identity.
---STEP 4: cosec x = 1 / (3/5).
---STEP 5: To divide by a fraction, we multiply by its reciprocal.
---STEP 6: cosec x = 1 * (5/3).
---STEP 7: cosec x = 5/3.
---ANSWER: The value of cosec x is 5/3.

Why It Matters

This identity helps engineers design structures and calculate forces in bridges and buildings. In AI/ML, understanding such reciprocal relationships can optimize algorithms for pattern recognition. Doctors use similar mathematical principles to understand wave patterns in medical imaging, helping them diagnose conditions.

Common Mistakes

MISTAKE: Thinking cosec x = sin x. | CORRECTION: Cosec x is the reciprocal of sin x, meaning cosec x = 1 / sin x, not equal to sin x.

MISTAKE: Forgetting that sin x cannot be 0 when calculating cosec x. | CORRECTION: Division by zero is undefined. If sin x = 0, then cosec x is undefined, just like you can't divide 1 by 0.

MISTAKE: Confusing cosec x with cos x or cot x. | CORRECTION: Cosec x is specifically the reciprocal of sin x. Cos x is a different ratio, and cot x is the reciprocal of tan x.

Practice Questions

Try It Yourself

QUESTION: If sin A = 1/2, find cosec A. | ANSWER: cosec A = 2

QUESTION: A right-angled triangle has an angle B where the opposite side is 8 units and the hypotenuse is 10 units. Find cosec B. | ANSWER: cosec B = 10/8 = 5/4

QUESTION: If 2 sin theta = 1, what is the value of cosec theta? | ANSWER: sin theta = 1/2, so cosec theta = 2

MCQ

Quick Quiz

Which of the following is true?

cosec x = sin x

cosec x = 1 / cos x

cosec x = 1 / sin x

cosec x = tan x

The Correct Answer Is:

The identity states that cosec x is the reciprocal of sin x, which means cosec x = 1 / sin x. Options A, B, and D represent incorrect relationships.

Real World Connection

In the Real World

This identity is crucial in understanding wave phenomena, like sound waves or radio waves used in mobile phones. When engineers design antennas for your smartphone to catch network signals, they use these trigonometric relationships to predict how waves will travel and interact, ensuring you get clear calls and fast internet data.

Key Vocabulary

Key Terms

RECIPROCAL: The value that, when multiplied by another value, gives 1. For example, the reciprocal of 2 is 1/2. | SINE (sin x): A trigonometric ratio in a right-angled triangle, defined as Opposite side / Hypotenuse. | COSECANT (cosec x): A trigonometric ratio in a right-angled triangle, defined as Hypotenuse / Opposite side. It is the reciprocal of sin x. | IDENTITY: An equation that is true for all possible values of its variables.

What's Next

What to Learn Next

Next, you should explore other reciprocal identities like 1 / cos x = sec x and 1 / tan x = cot x. Understanding all these will build a strong foundation for solving complex trigonometric problems and help you ace your exams!