Question

Help me with this math question​

Answer 1

`\text{Given that,} \sin x = \frac{\sqrt 5}3,~~ \text{and}~~~ \sin y =\frac{\sqrt 2}2\\\\\\~~~~~~~\sin x = \frac{\sqrt 5}3\\\\\implies \sin^2 x = \frac 59\\\\\implies 1- \cos^2 x = \frac 59\\\\\implies \cos^2 x =1- \frac59 \\\\\implies \cos^2 x = \frac 49\\\\\implies \cos x = \frac 23~~~~~~~;[\text{In quadrant I, all ratios are positive.}]\\\\\\\\`

`~~~~~~\sin y =\frac{√(2)}2\\ \\\implies \sin y = \frac 1{\sqrt 2}\\\\\implies \sin^2 y = \frac 12\\\\\implies 1-\cos^2 y = \frac 12\\\\\implies \cos^2 y = 1-\frac 12=\frac 12\\\\\implies \cos y = \frac 1{\sqrt 2}\\\\\\`

`\text{Now,}\\\\\cos(x+y) = \cos x \cos y - \sin x \sin y\\\\\\~~~~~~~~~~~~~~=\frac 23 \cdot \frac 1{\sqrt 2}- \frac{\sqrt 5}3 \cdot \frac 1{\sqrt 2}\\\\\\~~~~~~~~~~~~~~=(2- \sqrt 5)/(3\sqrt 2)\\\\\\~~~~~~~~~~~~~~=\frac{\sqrt2\left(2 -\sqrt 5 \right)}6`

Answer 2

x= 0.84106.. + 27n, r = 7 - 0.84106.. + 27n