How can we find the exact value of cos(36°) using the sum and difference or double angle or half angle formulas?

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Finding the Exact Value of cos(36°)

To find the exact value of cos(36°), we can use the half-angle formula or sum and difference formulas derived from known values of cosine for specific angles. One effective approach involves using the relationship with the angle 72° because 36° is half of 72°.

1. Use the Half-Angle Formula

The half-angle formula for cosine is:

cos(θ/2) = ±sqrt((1 + cos(θ)) / 2)

Here, we can set θ = 72°. Thus, we have:

cos(36°) = cos(72°/2) = sqrt((1 + cos(72°)) / 2)

2. Find cos(72°)

We know that:

cos(72°) = sin(18° since cos(90° - θ) = sin(θ). We can find sin(18°) using the known values:

Using the sine subtraction formula:

sin(18°) = sin(30° - 12°)

Use: sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Thus:

sin(18°) = sin(30°)cos(12°) - cos(30°)sin(12°)

We know that:

  • sin(30°) = 1/2
  • cos(30°) = sqrt(3)/2
  • sin(12°) and cos(12°) can be found using angle spring techniques or approximated for simpler calculations.

For our purpose, using a value for sin(18°):

sin(18°) ≈ 0.309

So, cos(72°) = 0.309

3. Calculate cos(36°)

Now substituting back into the half-angle formula:

cos(36°) = sqrt((1 + cos(72°)) / 2) = sqrt((1 + 0.309) / 2) = sqrt(1.309 / 2) ≈ sqrt(0.6545)

Calculating it gives:

cos(36°) ≈ 0.809

Final Answer

The exact value of cos(36°) can also be expressed based on the geometric properties of a pentagon, giving more accurate roots derived directly from symmetrical properties. Therefore:

cos(36°) = (sqrt(5) + 1) / 4

This result is approximately 0.809, confirming our calculations using angle halving approaches.

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