What is the Value of sin 90?

S6-SA2-0050

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 an angle approaches 90 degrees, the opposite side becomes almost equal to the hypotenuse.

Simple Example

Quick Example

Imagine a ladder leaning against a straight wall, like a ladder used to fix a light bulb in your house. If the ladder stands perfectly straight up against the wall, it forms a 90-degree angle with the ground. In this extreme case, the height it reaches on the wall is the same as the length of the ladder itself. This 'perfect match' represents sin 90 = 1.

Worked Example

Step-by-Step

Let's understand sin 90 using a unit circle, which is a circle with a radius of 1 unit.

Step 1: Draw a coordinate plane (x-axis and y-axis).
---Step 2: Draw a circle with its center at the origin (0,0) and a radius of 1 unit. This is our unit circle.
---Step 3: Start from the point (1,0) on the positive x-axis. This represents an angle of 0 degrees.
---Step 4: Rotate a point counter-clockwise along the circle until the angle formed with the positive x-axis is 90 degrees.
---Step 5: At 90 degrees, the point on the unit circle will be (0,1). This point is exactly on the positive y-axis.
---Step 6: For any point (x,y) on the unit circle, sin(angle) is equal to the y-coordinate. In our case, for 90 degrees, the y-coordinate is 1.
---Step 7: Therefore, sin 90 = 1.

Answer: The value of sin 90 is 1.

Why It Matters

Understanding sin 90 is crucial in many fields, from designing buildings to launching rockets. Engineers use it to calculate forces and angles in structures, ensuring they stand strong. Space scientists at ISRO use it in trajectory calculations for satellites, helping them reach space accurately.

Common Mistakes

MISTAKE: Thinking sin 90 is 0 or undefined. | CORRECTION: Remember that sin 90 is a specific, defined value and it is 1, not 0 or undefined.

MISTAKE: Confusing sin 90 with cos 90. | CORRECTION: While sin 90 is 1, cos 90 is 0. Always remember their distinct values.

MISTAKE: Assuming sin 90 is a different value in degrees versus radians. | CORRECTION: The value of sin(angle) is the same whether the angle is expressed as 90 degrees or pi/2 radians. The value 1 remains constant.

Practice Questions

Try It Yourself

QUESTION: What is the value of sin 90 degrees? | ANSWER: 1

QUESTION: If a ladder forms an angle of 90 degrees with the ground (meaning it's vertical), and its length is 5 meters, what is the height it reaches on the wall? (Hint: Think about sin 90) | ANSWER: 5 meters (because height = length * sin 90 = 5 * 1 = 5)

QUESTION: A robot arm is designed to lift a box. If the arm extends vertically upwards, making a 90-degree angle with its base, and its maximum reach is 1.2 meters, what is the vertical displacement from its base? | ANSWER: 1.2 meters (since vertical displacement = maximum reach * sin 90 = 1.2 * 1 = 1.2 meters)

MCQ

Quick Quiz

What is the numerical value of sin 90 degrees?

-1

Undefined

The Correct Answer Is:

The sine of 90 degrees is 1. This can be understood by considering the unit circle where at 90 degrees, the y-coordinate (which represents sine) is 1.

Real World Connection

In the Real World

In cricket, when a bowler bowls a 'full toss' straight up, it momentarily reaches its highest point directly above the bowler, creating a 90-degree angle with the horizontal. Understanding such angles helps sports analysts predict ball trajectories. Also, architects use trigonometry to ensure buildings stand perfectly straight and stable, where sin 90 often represents a perfectly vertical alignment.

Key Vocabulary

Key Terms

SINE: A trigonometric ratio of the opposite side to the hypotenuse in a right-angled triangle. | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle. | UNIT CIRCLE: A circle with a radius of 1, centered at the origin of a coordinate system. | TRIGONOMETRY: The branch of mathematics dealing with the relations between the sides and angles of triangles.

What's Next

What to Learn Next

Now that you know sin 90, explore the values of other special angles like sin 0, sin 30, sin 45, and sin 60. You can also learn about cos 90 and tan 90. These will help you build a strong foundation in trigonometry!