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Sin 45 Degrees Calculator

Reviewed by Calculator Editorial Team

This calculator helps you find the sine of 45 degrees. The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. Understanding sin(45°) is essential in geometry, physics, and engineering applications.

What is sin(45°)?

The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. For a 45-45-90 triangle, which is an isosceles right triangle, the two legs are of equal length, and the hypotenuse is √2 times the length of each leg.

When you calculate sin(45°), you're essentially finding the ratio of the length of the side opposite the 45° angle to the hypotenuse. This value is always the same for any 45-45-90 triangle, regardless of the triangle's size.

Key Formula

sin(θ) = opposite / hypotenuse

For θ = 45° in a right triangle with legs of length 1:

sin(45°) = 1 / √2 ≈ 0.7071

This value is an irrational number, often represented as √2/2 in exact form. The sine of 45 degrees is approximately 0.7071 when calculated to four decimal places.

How to calculate sin(45°)

Calculating sin(45°) is straightforward once you understand the basic trigonometric relationships. Here's a step-by-step method:

  1. Draw a right triangle with both non-right angles equal to 45°.
  2. Label the sides: let the two legs be of length 1 unit each.
  3. Use the Pythagorean theorem to find the hypotenuse: hypotenuse = √(1² + 1²) = √2.
  4. Apply the sine formula: sin(45°) = opposite / hypotenuse = 1 / √2.
  5. Simplify the expression: √2/2 ≈ 0.7071.

Important Note

The exact value of sin(45°) is √2/2, while the approximate decimal value is 0.7071. For most practical purposes, the decimal approximation is sufficient.

This method works for any 45-45-90 triangle, regardless of the actual lengths of the sides, because the ratios remain constant.

Example Calculation

Let's work through a concrete example to see how this calculation works in practice.

Example: Triangle with Legs of 5 cm

  1. Draw a right triangle with both non-right angles at 45°.
  2. Let each leg be 5 cm long.
  3. Calculate the hypotenuse: √(5² + 5²) = √(25 + 25) = √50 = 5√2 cm.
  4. Now calculate sin(45°): opposite / hypotenuse = 5 / (5√2) = 1/√2 ≈ 0.7071.

The result is the same as before, demonstrating that the sine of 45 degrees is independent of the triangle's size.

This example shows that no matter how large or small the triangle is, as long as it's a 45-45-90 triangle, the sine of 45 degrees will always be √2/2.

Visualizing sin(45°)

Understanding sin(45°) becomes even clearer when you visualize it on the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane.

At 45° from the positive x-axis, the coordinates of the point on the unit circle are (√2/2, √2/2). The y-coordinate of this point represents the sine of 45 degrees, which is √2/2.

Unit Circle Insight

The unit circle provides a geometric interpretation of trigonometric functions. For any angle θ, the point (cosθ, sinθ) lies on the unit circle.

This visualization helps in understanding the relationship between angles and their trigonometric values, which is fundamental in many areas of mathematics and science.

FAQ

What is the exact value of sin(45°)?

The exact value of sin(45°) is √2/2. This is an irrational number that cannot be expressed as a simple fraction.

How is sin(45°) different from sin(45 radians)?

Sin(45°) and sin(45 radians) are different because they use different units. 45° is 45 degrees, while 45 radians is approximately 2618.97 degrees. The sine function is periodic with a period of 2π radians (about 360°), so sin(45°) and sin(45 radians) will have different values.

Can sin(45°) be negative?

No, sin(45°) is always positive because 45° is in the first quadrant where all trigonometric functions are positive. The sine function becomes negative in the third and fourth quadrants.

What is the relationship between sin(45°) and cos(45°)?

For 45°, sin(45°) = cos(45°) = √2/2. This is because in a 45-45-90 triangle, the two legs are equal, making the sine and cosine values identical.