требуется помощь гения геометрии (задание б)
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Аноним:
напишіть свій інст, там допоможу з вирішенням
Ответы
Ответ дал:
1
The equation you've written is a trigonometric identity that holds true for any angle "a." It's a manifestation of the trigonometric identity known as the sum-to-product identity. Here's a breakdown of the identity you've written:
cos(a) + sin(a)sin(180° - a) = 1
The identity can be proven as follows:
1. Start with the sum-to-product identity for sine:
sin(A)sin(B) = 1/2[cos(A - B) - cos(A + B)]
2. Substitute A = a and B = (180° - a):
sin(a)sin(180° - a) = 1/2[cos(a - (180° - a)) - cos(a + (180° - a))]
3. Simplify the expressions inside the brackets:
cos(a - (180° - a)) = cos(a + a) = cos(2a)
cos(a + (180° - a)) = cos(a + a) = cos(2a)
4. Substitute these simplified expressions back into the identity:
1/2[cos(2a) - cos(2a)]
5. Since we are subtracting cos(2a) from itself, it equals zero:
1/2[0] = 0
6. Now, we have:
sin(a)sin(180° - a) = 0
7. Add cos(a) to both sides of the equation:
cos(a) + sin(a)sin(180° - a) = cos(a) + 0 = cos(a) = 1
So, the trigonometric identity cos(a) + sin(a)sin(180° - a) = 1 is true for any angle "a."
cos(a) + sin(a)sin(180° - a) = 1
The identity can be proven as follows:
1. Start with the sum-to-product identity for sine:
sin(A)sin(B) = 1/2[cos(A - B) - cos(A + B)]
2. Substitute A = a and B = (180° - a):
sin(a)sin(180° - a) = 1/2[cos(a - (180° - a)) - cos(a + (180° - a))]
3. Simplify the expressions inside the brackets:
cos(a - (180° - a)) = cos(a + a) = cos(2a)
cos(a + (180° - a)) = cos(a + a) = cos(2a)
4. Substitute these simplified expressions back into the identity:
1/2[cos(2a) - cos(2a)]
5. Since we are subtracting cos(2a) from itself, it equals zero:
1/2[0] = 0
6. Now, we have:
sin(a)sin(180° - a) = 0
7. Add cos(a) to both sides of the equation:
cos(a) + sin(a)sin(180° - a) = cos(a) + 0 = cos(a) = 1
So, the trigonometric identity cos(a) + sin(a)sin(180° - a) = 1 is true for any angle "a."
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