What is cos 45 degrees? | Simple Explanation for Class 10

S6-SA2-0293

What is the Value of cos 45 degrees?

Grade Level:

Class 10

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

Definition

What is it?

The value of cos 45 degrees is a specific ratio in a right-angled triangle. It represents the length of the adjacent side divided by the length of the hypotenuse when one of the acute angles is 45 degrees.

Simple Example

Quick Example

Imagine a cricket pitch where the boundary rope forms a perfect square corner (90 degrees). If you draw a straight line from the batsman's crease to the corner of the boundary, this line makes a 45-degree angle with the side of the pitch. The 'cos 45' value helps us understand the relationship between the length of the pitch side and the diagonal path.

Worked Example

Step-by-Step

Let's find the value of cos 45 degrees using a right-angled isosceles triangle. --- Step 1: Draw a right-angled triangle ABC, with the right angle at B. --- Step 2: Since it's an isosceles right-angled triangle, the two non-right angles are equal. So, Angle A = Angle C = 45 degrees. --- Step 3: Let the equal sides be AB = BC = 'a' units. --- Step 4: Use the Pythagorean theorem to find the hypotenuse AC. AC^2 = AB^2 + BC^2 = a^2 + a^2 = 2a^2. So, AC = sqrt(2a^2) = a * sqrt(2). --- Step 5: Recall the definition of cosine: cos(angle) = (Adjacent Side) / (Hypotenuse). --- Step 6: For angle A (45 degrees), the adjacent side is AB = 'a' and the hypotenuse is AC = a * sqrt(2). --- Step 7: Therefore, cos 45 degrees = a / (a * sqrt(2)) = 1 / sqrt(2). --- Answer: The value of cos 45 degrees is 1 / sqrt(2).

Why It Matters

Understanding trigonometric values like cos 45 is crucial in fields like AI/ML for image processing and robotics to calculate angles and distances. Engineers use it to design bridges and buildings, while physicists apply it to understand wave motion and projectile trajectories. It's a fundamental tool for solving real-world problems.

Common Mistakes

MISTAKE: Remembering cos 45 as sqrt(2) | CORRECTION: The value of cos 45 degrees is 1 / sqrt(2), not just sqrt(2). Always remember it's a fraction.

MISTAKE: Confusing cos 45 with sin 45 | CORRECTION: While cos 45 and sin 45 both have the value 1 / sqrt(2), it's important to know their definitions: cos = Adjacent/Hypotenuse and sin = Opposite/Hypotenuse. They are equal only for 45 degrees.

MISTAKE: Forgetting to rationalize the denominator | CORRECTION: Often, 1 / sqrt(2) is written as sqrt(2) / 2 after multiplying the numerator and denominator by sqrt(2). Both are correct, but sqrt(2) / 2 is generally preferred in final answers.

Practice Questions

Try It Yourself

QUESTION: What is the exact numerical value of cos 45 degrees? | ANSWER: 1 / sqrt(2)

QUESTION: If a ladder leans against a wall making an angle of 45 degrees with the ground, and the base of the ladder is 3 meters from the wall, what is the length of the ladder? (Hint: cos 45 = Adjacent / Hypotenuse) | ANSWER: Length of ladder = 3 * sqrt(2) meters

QUESTION: A drone flies at a constant height. If it is 100 meters away horizontally from its launch point and the angle of elevation from the launch point to the drone is 45 degrees, how far is the drone from the launch point in a straight line (hypotenuse distance)? | ANSWER: 100 * sqrt(2) meters

MCQ

Quick Quiz

Which of the following is the correct value for cos 45 degrees?

sqrt(2)

2026-01-02T00:00:00.000Z

1/sqrt(2)

sqrt(3)/2

The Correct Answer Is:

The correct value for cos 45 degrees is 1/sqrt(2). This is a standard trigonometric value derived from a 45-45-90 degree right-angled triangle. Options A, B, and D are incorrect values.

Real World Connection

In the Real World

In India, ISRO scientists use trigonometry to calculate the trajectories of rockets and satellites. When a satellite's path is tracked, angles like 45 degrees are critical for determining its position and speed. Similarly, civil engineers designing flyovers in cities like Bengaluru or Mumbai use these values to ensure the structural stability and precise angles of ramps.

Key Vocabulary

Key Terms

COSINE: Ratio of the adjacent side to the hypotenuse in a right-angled triangle | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle | ADJACENT SIDE: The side next to the angle being considered, not the hypotenuse | RATIONALIZE: To remove square roots from the denominator of a fraction

What's Next

What to Learn Next

Now that you understand cos 45, try learning about sin 45 and tan 45 degrees. You'll see that sin 45 is also 1/sqrt(2), which is an interesting property for this angle. Then, you can move on to other common angles like 30 and 60 degrees!