What is the Value of cos 90?

S6-SA2-0051

Grade Level:

Class 10

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

Definition

What is it?

The value of cos 90 degrees is a fundamental concept in trigonometry, representing the cosine of an angle of 90 degrees. In a right-angled triangle, if one of the acute angles is considered, its cosine value is the ratio of the length of the adjacent side to the length of the hypotenuse. For a 90-degree angle, this value is 0.

Simple Example

Quick Example

Imagine you are flying a kite directly above your head. The string goes straight up, making a 90-degree angle with the ground where you are standing. If we think about the 'adjacent side' in this situation, it would be the distance along the ground from you to the point directly below the kite. Since the kite is directly above you, this distance is zero. So, cos 90 is 0, just like that ground distance!

Worked Example

Step-by-Step

Let's find the value of cos 90 using the unit circle concept:
1. Imagine a circle with a radius of 1 unit centered at the origin (0,0) of a coordinate plane.
2. Start from the positive x-axis, which represents 0 degrees.
3. Rotate counter-clockwise by 90 degrees.
4. You will reach the point (0, 1) on the positive y-axis.
5. In the unit circle, for any angle 'theta', the x-coordinate of the point where the angle's terminal side intersects the circle is cos(theta).
6. At 90 degrees, the point is (0, 1).
7. The x-coordinate of this point is 0.
8. Therefore, cos 90 = 0.

Why It Matters

Understanding cos 90 is crucial in many fields like Physics for calculating forces and motion, especially when things are perpendicular. Engineers use it to design structures and machines, ensuring stability and efficiency. In AI/ML, it helps define angles in data analysis, which is vital for developing smart algorithms.

Common Mistakes

MISTAKE: Confusing cos 90 with sin 90 or tan 90. | CORRECTION: Remember that cos 90 = 0, while sin 90 = 1 and tan 90 is undefined. Each trigonometric ratio has a unique value at 90 degrees.

MISTAKE: Thinking that cos 90 is 1 because 90 is a 'full' quarter turn. | CORRECTION: Cosine represents the x-coordinate on the unit circle. At 90 degrees, the point is on the y-axis, meaning its x-coordinate is 0, not 1.

MISTAKE: Forgetting the exact value and trying to calculate it using a calculator for simple problems. | CORRECTION: Memorize the values for common angles (0, 30, 45, 60, 90 degrees) for sin, cos, and tan. This speeds up problem-solving and builds a stronger foundation.

Practice Questions

Try It Yourself

QUESTION: What is the value of cos 90 degrees? | ANSWER: 0

QUESTION: If a ladder is placed vertically against a wall, making an angle of 90 degrees with the ground, what is the cosine of this angle? | ANSWER: 0

QUESTION: Evaluate: 5 * cos 90 + sin 0 + tan 45. | ANSWER: 5 * 0 + 0 + 1 = 1

MCQ

Quick Quiz

What is the numerical value of cos 90 degrees?

-1

Undefined

The Correct Answer Is:

The cosine of 90 degrees is 0. This can be understood using the unit circle where the x-coordinate at 90 degrees is 0. Options A, C, and D are incorrect values for cos 90.

Real World Connection

In the Real World

When ISRO launches a rocket straight up, its initial trajectory makes an angle close to 90 degrees with the horizontal ground. While the rocket doesn't stay at 90 degrees, understanding the forces acting at various angles, including 90 degrees, is crucial for calculating thrust and stability. This involves trigonometry, where cos 90 plays a role in components of forces.

Key Vocabulary

Key Terms

TRIGONOMETRY: The branch of mathematics dealing with the relations of the sides and angles of triangles. | COSINE: A trigonometric ratio in a right-angled triangle, defined as the ratio of the length of the adjacent side to the length of the hypotenuse. | UNIT CIRCLE: A circle with a radius of one unit, used to understand trigonometric functions. | ANGLE: The amount of turn between two lines meeting at a common point.

What's Next

What to Learn Next

Now that you know the value of cos 90, you should explore the values of sin 90 and tan 90. Then, move on to understanding trigonometric values for other special angles like 0, 30, 45, and 60 degrees. This will build a strong foundation for solving more complex trigonometry problems!