Question

Sin^5a-cos^5a=? если sina-cosa=1/2​

Answer 1

Возводим левую и правую части равенства:

$1-\sin2\alpha=\frac{1}{4}~~~~\Leftrightarrow~~~ \sin2\alpha=\frac{3}{4}$

$\sin^5\alpha-\cos^5\alpha=(\sin\alpha-\cos\alpha)(\sin^4\alpha+\sin^3\alpha\cos\alpha+\sin^2\alpha\cos^2\alpha+\\ \\ +\sin\alpha\cos^3\alpha+\cos^4\alpha)=\frac{1}{2}((\sin^2\alpha+\cos^2\alpha)^2-\sin2\alpha+\frac{1}{2}\sin^2\alpha\sin2\alpha+\\ \\ +\frac{1}{4}\sin^22\alpha+\frac{1}{2}\cos^2\alpha\sin2\alpha)=\frac{1}{2}(1-\sin2\alpha+\frac{1}{2}\sin2\alpha+\frac{1}{4}\sin^22\alpha)=\\ \\ \\ =\frac{1}{2}(1-\frac{1}{2}\cdot\frac{3}{4}+\frac{1}{4}\cdot\frac{9}{16})=\frac{49}{128}$

Answer 2

To find the value of sin^5(a) - cos^5(a), given that sin(a) - cos(a) = 1/2, we can use the trigonometric identity:

sin^2(a) - cos^2(a) = (sin(a) - cos(a))(sin(a) + cos(a))

Squaring both sides of the equation sin(a) - cos(a) = 1/2:

(sin(a) - cos(a))^2 = (1/2)^2 sin^2(a) - 2sin(a)cos(a) + cos^2(a) = 1/4

Now, using the given value of sin(a) - cos(a) = 1/2, we can substitute it into the equation:

(1/2)^2 - 2sin(a)cos(a) = 1/4 1/4 - 2sin(a)cos(a) = 1/4 -2sin(a)cos(a) = 0

Since sin(a) and cos(a) cannot both be zero (since they are not equal), the only solution is sin(a)cos(a) = 0.

Now, let's find the value of sin^5(a) - cos^5(a):

sin^5(a) - cos^5(a) = (sin^3(a) + sin^2(a)cos(a) + sin(a)cos^2(a) + cos^3(a))(sin(a) - cos(a))

Since we know sin(a)cos(a) = 0, the above equation simplifies to:

sin^5(a) - cos^5(a) = (sin^3(a) + cos^3(a))(sin(a) - cos(a))

Now, we can use the sum of cubes formula:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Let a = sin(a) and b = cos(a):

sin^3(a) + cos^3(a) = (sin(a) + cos(a))(sin^2(a) - sin(a)cos(a) + cos^2(a))

Since sin(a)cos(a) = 0:

sin^3(a) + cos^3(a) = (sin(a) + cos(a))(sin^2(a) + cos^2(a))

And, since sin^2(a) + cos^2(a) = 1 (the Pythagorean identity):

sin^3(a) + cos^3(a) = (sin(a) + cos(a))(1)

Finally, substituting sin(a) - cos(a) = 1/2:

sin^5(a) - cos^5(a) = (1/2)(1)

sin^5(a) - cos^5(a) = 1/2

So, the value of sin^5(a) - cos^5(a) is 1/2.