What is the Value of sin 45?

S6-SA2-0044

Grade Level:

Class 10

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

Definition

What is it?

The value of sin 45 degrees is a fundamental trigonometric ratio that describes the relationship between the side opposite to a 45-degree angle and the hypotenuse in a right-angled triangle. It is a constant value, always 1/sqrt(2) or approximately 0.707.

Simple Example

Quick Example

Imagine you are flying a kite and the string makes a 45-degree angle with the ground. If the kite string (hypotenuse) is 100 meters long, the height of the kite from the ground (opposite side) would be 100 * sin 45. Knowing sin 45 helps you quickly find the kite's height without needing to measure it directly.

Worked Example

Step-by-Step

Let's find the value of sin 45 degrees using an isosceles right-angled triangle. --- Step 1: Draw a right-angled triangle ABC, where angle B is 90 degrees. --- Step 2: Since it's an isosceles right-angled triangle, angles A and C must be equal. So, Angle A = Angle C = (180 - 90)/2 = 45 degrees. --- Step 3: Let the equal sides AB and BC be 'a' units long. --- Step 4: Use the Pythagoras 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 sin (angle) = Opposite side / Hypotenuse. --- Step 6: For angle A (45 degrees), the opposite side is BC = a, and the hypotenuse is AC = a * sqrt(2). --- Step 7: Therefore, sin 45 = BC / AC = a / (a * sqrt(2)) = 1 / sqrt(2). --- The value of sin 45 degrees is 1/sqrt(2).

Why It Matters

Understanding sin 45 is crucial for engineers designing bridges or buildings, as it helps calculate forces and angles. In game development, it's used to make characters move realistically. Future doctors might use it in medical imaging, and space scientists at ISRO use it for satellite trajectory calculations.

Common Mistakes

MISTAKE: Confusing sin 45 with cos 45. | CORRECTION: Remember that sin 45 = 1/sqrt(2) and cos 45 = 1/sqrt(2). For 45 degrees, both sine and cosine values are the same, but for other angles, they are different. Always double-check the ratio (Opposite/Hypotenuse for sin, Adjacent/Hypotenuse for cos).

MISTAKE: Forgetting to rationalize the denominator, leaving the answer as 1/sqrt(2) instead of sqrt(2)/2. | CORRECTION: While 1/sqrt(2) is correct, it's good practice to rationalize the denominator by multiplying both numerator and denominator by sqrt(2) to get (1 * sqrt(2)) / (sqrt(2) * sqrt(2)) = sqrt(2)/2.

MISTAKE: Assuming sin 45 is 0.5 or 1. | CORRECTION: The value of sin 45 is 1/sqrt(2) which is approximately 0.707. Remember that sine values range from 0 to 1 for angles from 0 to 90 degrees.

Practice Questions

Try It Yourself

QUESTION: What is the approximate decimal value of sin 45, rounded to two decimal places? | ANSWER: 0.71

QUESTION: If the hypotenuse of a right-angled triangle is 8 units and one of its angles is 45 degrees, what is the length of the side opposite to the 45-degree angle? | ANSWER: 8 * (1/sqrt(2)) = 8/sqrt(2) = 4 * sqrt(2) units.

QUESTION: In an isosceles right-angled triangle, if the two equal sides are each 5 cm long, what is the value of sin of one of the acute angles? | ANSWER: The hypotenuse is sqrt(5^2 + 5^2) = sqrt(50) = 5 * sqrt(2) cm. sin 45 = Opposite / Hypotenuse = 5 / (5 * sqrt(2)) = 1/sqrt(2).

MCQ

Quick Quiz

Which of the following is the correct value of sin 45 degrees?

2026-01-02T00:00:00.000Z

1/sqrt(2)

sqrt(3)/2

The Correct Answer Is:

The value of sin 45 degrees is 1/sqrt(2). Options A, B, and D correspond to sin 0, sin 30, and sin 60 degrees, respectively, or are incorrect.

Real World Connection

In the Real World

When you see construction workers using a ladder, they often ensure it's placed at a safe angle. If a ladder needs to reach a height of 5 meters and is placed at a 45-degree angle, they use sin 45 to calculate the exact length of the ladder needed. This ensures stability and safety on Indian construction sites.

Key Vocabulary

Key Terms

TRIGONOMETRY: A branch of mathematics dealing with the relationships between the sides and angles of triangles. | SINE: A trigonometric ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle. | RIGHT-ANGLED TRIANGLE: A triangle with one angle measuring 90 degrees. | RATIONALIZE DENOMINATOR: The process of removing a radical (like a square root) from the denominator of a fraction.

What's Next

What to Learn Next

Now that you've mastered sin 45, you're ready to explore the values of cos 45 and tan 45. These concepts build directly on what you've learned and will complete your understanding of trigonometric ratios for 45 degrees, opening doors to solving more complex triangle problems!