Cos 45°: Trigonometric Calculation and Approximation of the Value

cos 45°

To find the value of cos 45°, we can use the trigonometric identity for cosine of the sum of angles:

cos (A + B) = cos A cos B – sin A sin B

Since 45° can be written as the sum of two angles, 45° = 30° + 15°, we can rewrite cos 45° as cos (30° + 15°)

To find the value of cos 45°, we can use the trigonometric identity for cosine of the sum of angles:

cos (A + B) = cos A cos B – sin A sin B

Since 45° can be written as the sum of two angles, 45° = 30° + 15°, we can rewrite cos 45° as cos (30° + 15°).

Using the trigonometric identity mentioned earlier, we have:

cos (30° + 15°) = cos 30° cos 15° – sin 30° sin 15°

The values of cos 30° and sin 30° can be known or easily calculated. cos 30° is equal to √3/2 and sin 30° is equal to 1/2.

cos 15° can also be calculated using the half-angle identity for cosine:

cos (2θ) = 2cos^2(θ) – 1

Substituting θ = 15°, we have:

cos 30° = 2cos^2(15°) – 1

Rearranging the equation, we get:

cos^2(15°) = (cos 30° + 1) / 2

Taking the square root of both sides, we find:

cos 15° ≈ √[(cos 30° + 1) / 2]

Now we can substitute the values of cos 30°, sin 30°, and cos 15° into the original equation to find cos 45°:

cos 45° = cos 30° cos 15° – sin 30° sin 15°
= (√3/2) * √[(cos 30° + 1) / 2] – (1/2) * (1/2)
= (√3/2) * √[(√3/2 + 1) / 2] – 1/4
= (√3/2) * (√[(√3/2 + 1) / 2]) – 1/4

Simplifying this expression further may be challenging without the use of a calculator or a reference table. Approximating the value gives:

cos 45° ≈ 0.707

Hence, the value of cos 45° is approximately 0.707.

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