Cos 45 value

Mathematics
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cos 45 value

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What is the value of \cos 45^\circ?

Answer:
The value of \cos 45^\circ can be derived using the properties of a 45-45-90 triangle, which is a special right triangle. Let’s explore how to find the cosine of 45^\circ step-by-step.

  1. Understanding the 45-45-90 Triangle:

    • A 45-45-90 triangle has two 45-degree angles and one 90-degree angle.
    • The lengths of the legs of this triangle are equal.
    • If each leg has a length of 1, then you can use the Pythagorean theorem to find the length of the hypotenuse.

  2. Using the Pythagorean Theorem:

    • Let the lengths of the legs be a and b and the hypotenuse be c.

      For a 45-45-90 triangle:

      a = b = 1
      c = \sqrt{a^2 + b^2} = \sqrt{1^2 + 1^2} = \sqrt{2}

  3. Calculating \cos 45^\circ:

    • The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

      For \cos 45^\circ:

      \cos 45^\circ = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}

  4. Simplifying the Value:

    • To express the result in a more standard form, rationalize the denominator:

      \cos 45^\circ = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}

Final Answer:
The value of \cos 45^\circ is

\boxed{\frac{\sqrt{2}}{2}}