Question

Sin^2(72°)+sin^2(18°)

Answer 1

Ответ:

1

Пошаговое объяснение:

sin(90°-18°)=cos(18°)

sin(72°)=cos(18°)

cos^2(18°)+sin^2(18°)=1.

Answer 2

To calculate the value of sin^2(72°) + sin^2(18°), we'll use the trigonometric identity:

sin^2(x) + cos^2(x) = 1

Rearranging this equation, we get:

sin^2(x) = 1 - cos^2(x)

Now let's substitute the given angles:

sin^2(72°) = 1 - cos^2(72°) sin^2(18°) = 1 - cos^2(18°)

To find the values of cos(72°) and cos(18°), we can use the fact that:

cos(90° - x) = sin(x)

Therefore:

cos(72°) = sin(18°) cos(18°) = sin(72°)

Now let's substitute these values into the equations:

sin^2(72°) = 1 - cos^2(72°) = 1 - sin^2(18°) sin^2(18°) = 1 - cos^2(18°) = 1 - sin^2(72°)

Substituting sin^2(72°) = 1 - sin^2(18°) into the second equation, we get:

sin^2(18°) = 1 - cos^2(18°) = 1 - (1 - sin^2(72°)) = sin^2(72°)

Therefore, sin^2(72°) + sin^2(18°) = sin^2(72°) + sin^2(72°) = 2 sin^2(72°).

To simplify this further, we can use the identity:

sin(2x) = 2 sin(x) cos(x)

Let's substitute x = 72°:

sin(144°) = 2 sin(72°) cos(72°)

Since sin(144°) = sin(180° - 144°) = sin(36°), we have:

sin^2(36°) = 2 sin^2(72°)

Therefore, sin^2(72°) + sin^2(18°) = 2 sin^2(72°) = sin^2(36°).