What is the Value of sin 90 degrees?

S6-SA2-0298

Grade Level:

Class 10

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

Definition

What is it?

The value of sin 90 degrees is 1. In a right-angled triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. When the angle is 90 degrees, this ratio reaches its maximum possible value.

Simple Example

Quick Example

Imagine a ladder leaning against a wall. If the ladder is perfectly upright, making a 90-degree angle with the ground, and you want to calculate how high it reaches compared to its own length, that's like finding sin 90. Since it's standing straight, the height it reaches is exactly its full length, so the ratio is 1.

Worked Example

Step-by-Step

Let's find the value of sin 90 degrees using a unit circle. --- Step 1: Draw a circle with a radius of 1 unit centered at the origin (0,0) on a graph. This is called a unit circle. --- Step 2: Start from the positive x-axis (0 degrees). Rotate counter-clockwise to an angle of 90 degrees. --- Step 3: At 90 degrees, the point on the unit circle will be exactly at (0, 1). --- Step 4: In a unit circle, for any angle theta, the coordinates of the point on the circle are (cos theta, sin theta). --- Step 5: So, for 90 degrees, the coordinates are (cos 90, sin 90), which is (0, 1). --- Step 6: This means cos 90 = 0 and sin 90 = 1. --- Answer: The value of sin 90 degrees is 1.

Why It Matters

Understanding sin 90 is crucial in many advanced fields. Engineers use it to design structures that stand perfectly upright, while physicists use it to calculate forces acting at right angles. It's fundamental for careers in architecture, robotics, and even designing the launch trajectories for rockets at ISRO.

Common Mistakes

MISTAKE: Thinking sin 90 is 0. | CORRECTION: sin 0 degrees is 0. sin 90 degrees is 1. Remember the graph of the sine function starts at 0, goes up to 1 at 90 degrees, and then back down.

MISTAKE: Confusing sin 90 with cos 90. | CORRECTION: sin 90 = 1, but cos 90 = 0. They are different values at 90 degrees.

MISTAKE: Assuming sin 90 has no real-world application. | CORRECTION: It's used everywhere, from designing straight walls in buildings to understanding how light waves behave when perfectly perpendicular.

Practice Questions

Try It Yourself

QUESTION: What is the value of 2 * sin 90 degrees? | ANSWER: 2

QUESTION: If sin x = 1, and x is an acute angle, what is the value of x? | ANSWER: 90 degrees

QUESTION: A right-angled triangle has its hypotenuse as 10 cm. If one of the acute angles approaches 90 degrees, what would be the approximate length of the side opposite to that angle? | ANSWER: 10 cm (Because sin 90 = Opposite/Hypotenuse = 1, so Opposite = Hypotenuse)

MCQ

Quick Quiz

What is the numerical value of sin 90 degrees?

0.5

Undefined

The Correct Answer Is:

The sine function represents the y-coordinate on a unit circle. At 90 degrees, the point on the unit circle is (0, 1), so sin 90 degrees is 1. Options A, C, and D are incorrect.

Real World Connection

In the Real World

When a crane lifts a heavy object, the cable often hangs perfectly vertical, making a 90-degree angle with the horizontal ground. The tension in the cable, and how effectively it lifts, uses the concept of sin 90 degrees to calculate the maximum upward force being applied. This is crucial for safety and efficiency in construction sites across India.

Key Vocabulary

Key Terms

SINE: A trigonometric ratio in a right-angled triangle, defined as the ratio of the length of the side opposite an acute angle to the length of the hypotenuse. | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle. | UNIT CIRCLE: A circle with a radius of 1 unit, centered at the origin of a coordinate system, used to understand trigonometric functions. | ANGLE: The amount of turn between two lines or surfaces meeting at a common point.

What's Next

What to Learn Next

Now that you understand sin 90, you should explore the values of cos 90 and tan 90. These concepts are interconnected and will help you build a stronger foundation in trigonometry, preparing you for more complex problems in Class 11 and 12.