What is the Range of sin x?

S6-SA2-0356

Grade Level:

Class 10

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

Definition

What is it?

The range of sin x refers to all the possible output values that the sine function can produce. For any angle x, the value of sin x will always fall between -1 and 1, inclusive.

Simple Example

Quick Example

Imagine you're tracking the temperature in your city over a year. If the temperature never goes below 5 degrees Celsius and never above 40 degrees Celsius, then the 'range' of temperature is from 5 to 40. Similarly, for sin x, its 'temperature' (value) always stays between -1 and 1.

Worked Example

Step-by-Step

Let's find the range of sin x for different angles.

Step 1: Consider sin(0 degrees). Its value is 0.
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Step 2: Consider sin(30 degrees). Its value is 0.5.
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Step 3: Consider sin(90 degrees). Its value is 1. This is the maximum value.
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Step 4: Consider sin(180 degrees). Its value is 0.
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Step 5: Consider sin(270 degrees). Its value is -1. This is the minimum value.
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Step 6: Consider sin(360 degrees). Its value is 0.
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Answer: As you can see, no matter the angle, the values always stay between -1 and 1. So, the range of sin x is [-1, 1].

Why It Matters

Understanding the range of sin x is super important in fields like Physics to model waves (like sound or light) and in Engineering to design circuits. It's also used in AI/ML for certain types of data processing and even in Space Technology to calculate satellite orbits.

Common Mistakes

MISTAKE: Thinking the range of sin x is from 0 to 1. | CORRECTION: Remember that sine values can also be negative, going down to -1.

MISTAKE: Confusing range with domain. | CORRECTION: Domain refers to the possible input angles (x), while range refers to the possible output values (sin x).

MISTAKE: Forgetting that the maximum and minimum values (1 and -1) are included in the range. | CORRECTION: The range is a closed interval, meaning 1 and -1 are part of the possible values, written as [-1, 1].

Practice Questions

Try It Yourself

QUESTION: Can sin x ever be 1.5? | ANSWER: No, because 1.5 is outside the range [-1, 1].

QUESTION: If sin x = -0.8, is this a valid value? | ANSWER: Yes, because -0.8 is within the range [-1, 1].

QUESTION: What is the range of 2 * sin x? | ANSWER: The range would be [-2, 2], because multiplying sin x by 2 doubles its maximum and minimum values.

MCQ

Quick Quiz

Which of the following values can sin x NOT take?

0.7

-0.3

1.2

The Correct Answer Is:

The range of sin x is from -1 to 1, inclusive. 0.7, -0.3, and 1 all fall within this range. 1.2 is greater than 1, so sin x cannot take this value.

Real World Connection

In the Real World

When engineers design earthquake-resistant buildings, they use sine functions to model how buildings might sway. The 'range' of the sine function helps them understand the maximum and minimum displacement, ensuring the structure can handle the movement without breaking. Similarly, in your mobile phone, the signals use waves whose 'strength' or 'amplitude' (related to range) is carefully controlled.

Key Vocabulary

Key Terms

RANGE: The set of all possible output values of a function | SINE FUNCTION: A trigonometric function that relates an angle of a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse | INTERVAL: A set of real numbers that includes all numbers between two specified numbers | TRIGONOMETRY: The branch of mathematics dealing with the relations of the sides and angles of triangles

What's Next

What to Learn Next

Great job understanding the range of sin x! Next, you should explore the range of cos x and tan x. These concepts build on what you've learned here and are crucial for solving more complex trigonometry problems.