Question

Дам 100 БАЛЛОВ, пж решите

Answer 1

8.

$\sin( \alpha ) \cos( \alpha ) = 0.2 \\ \sin( \alpha ) + \cos( \alpha ) = \: ? \\ \\ ( \sin( \alpha ) + \cos( \alpha ) ) {}^{2} = \sin {}^{2} ( \alpha ) + 2 \sin( \alpha ) \cos {}^{} ( \alpha ) + \cos {}^{2} ( \alpha ) \\ = 1 + 2 \sin( \alpha ) \cos( \alpha ) = 1 + 2 \times 0.2 = 1 + 0.4 = 1.4 \\ 1) \: \sin( \alpha ) + \cos( \alpha ) = - \sqrt{1.4} \\ 2) \: \sin( \alpha ) + \cos( \alpha ) = \sqrt{1.4}$

9.

$\sin( \alpha ) - \cos( \alpha ) = 0.7 \\ \sin {}^{4} ( \alpha ) - \cos {}^{4} ( \alpha ) = \: ? \\ \\ \sin {}^{4} ( \alpha ) - \cos {}^{4} ( \alpha ) = ( \sin {}^{2} ( \alpha ) - \cos {}^{2} ( \alpha ) )( \sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) ) = \\ = \sin {}^{2} ( \alpha ) - \cos {}^{2} ( \alpha ) = ( \sin( \alpha ) - \cos( \alpha ) )( \sin( \alpha ) + \cos( \alpha ) ) = \\ = 0.7( \sin( \alpha ) + \cos( \alpha ) )\\ \\ ( \sin( \alpha ) - \cos( \alpha ) ) {}^{2} = \sin {}^{2} ( \alpha ) - 2 \sin( \alpha ) \cos( \alpha ) + \cos {}^{2} ( \alpha ) = \\0.49 = 1 - 2 \sin( \alpha ) \cos( \alpha ) \\ 2 \sin( \alpha ) \cos( \alpha ) = 1 - 0.49 \\ 2 \sin( \alpha ) \cos(a) = 0.51 \\ \\ (\sin( \alpha ) + \cos( \alpha ) ) {}^{2} = \sin {}^{2} ( \alpha ) + 2 \sin( \alpha ) \cos( \alpha ) + \cos {}^{2} ( \alpha ) = \\ = 1 + 2 \sin( \alpha ) \cos( \alpha ) = 1 + 0.51= 1.51 \\ \\ 1) \: \sin( \alpha ) + \cos( \alpha ) = \sqrt{1.51} \\ \sin {}^{4} ( \alpha ) - \cos {}^{4} ( \alpha ) = 0.7 \times \sqrt{1.51} \\ \\ 2) \: \sin( \alpha ) + \cos( \alpha ) = - \sqrt{1.51} \\ \sin {}^{4} ( \alpha ) - \cos {}^{4} ( \alpha ) = 0.7 \times ( - \sqrt{1.51} ) = - 0.7 \times \sqrt{1.51}$

10.

$\cos( \alpha ) \sin( \alpha ) = 0.4 \\ 2\cos( \alpha ) \sin( \alpha ) = 0.8 \\ \cos {}^{3} ( \alpha ) + \sin {}^{3} ( \alpha ) = \: ? \\ \\ \cos {}^{3} ( \alpha ) + \sin {}^{3} ( \alpha ) = ( \sin {}^{} ( \alpha ) + \cos {}^{} ( \alpha ) ) ( \sin {}^{2} ( \alpha ) - \sin( \alpha ) \cos( \alpha ) + \cos {}^{2} ( \alpha )) = \\ = ( \sin( \alpha ) + \cos( \alpha ) )(1 - 0.4) = 0.6( \sin( \alpha ) + \cos( \alpha ) )\\ \\ ( \sin( \alpha ) + \cos( \alpha ) ) {}^{2} = \sin {}^{2} ( \alpha ) + 2 \sin( \alpha ) \cos( \alpha ) + \cos {}^{2} ( \alpha ) = \\ = 1 + 0.8 = 1.8 \\ \\ 1) \: \sin( \alpha ) + \cos( \alpha ) = \sqrt{1.8} \\ \cos {}^{3} ( \alpha ) + \sin {}^{3} ( \alpha ) = 0.6 \times \sqrt{1.8} \\ \\ 2) \: \sin( \alpha ) + \cos( \alpha ) = - \sqrt{1.8} \\ \cos {}^{3} ( \alpha ) + \sin {}^{3} ( \alpha ) = 0.6 \times ( - \sqrt{1.8} ) = - 0.6 \times \sqrt{1.8}$