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To solve the problem number 3: 

Given:
\[ \sin x = \frac{1}{\sqrt{5}} \]
\[ \cos y = \frac{2}{3} \]
Calculate \( K = \tan(x + y) \).

First, we need to find the values of \(\cos x\) and \(\sin y\).

For \(\sin x = \frac{1}{\sqrt{5}}\):
\[ \cos x = \sqrt{1 - \sin^2 x} = \sqrt{1 - \left(\frac{1}{\sqrt{5}}\right)^2} = \sqrt{1 - \frac{1}{5}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \]

For \(\cos y = \frac{2}{3}\):
\[ \sin y = \sqrt{1 - \cos^2 y} = \sqrt{1 - \left(\frac{2}{3}\right)^2} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \]

Next, we use the angle addition formula for tangent:
\[ \tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \]

First, we calculate \(\tan x\) and \(\tan y\):
\[ \tan x = \frac{\sin x}{\cos x} = \frac{\frac{1}{\sqrt{5}}}{\frac{2}{\sqrt{5}}} = \frac{1}{2} \]
\[ \tan y = \frac{\sin y}{\cos y} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{2} \]

Now substitute \(\tan x\) and \(\tan y\) into the formula:
\[ \tan(x + y) = \frac{\frac{1}{2} + \frac{\sqrt{5}}{2}}{1 - \left(\frac{1}{2}\right) \left(\frac{\sqrt{5}}{2}\right)} = \frac{\frac{1 + \sqrt{5}}{2}}{1 - \frac{\sqrt{5}}{4}} = \frac{1 + \sqrt{5}}{2 - \frac{\sqrt{5}}{2}} = \frac{1 + \sqrt{5}}{\frac{4 - \sqrt{5}}{2}} = \frac{2(1 + \sqrt{5})}{4 - \sqrt{5}} = \frac{2 + 2\sqrt{5}}{4 - \sqrt{5}} \]

To rationalize the denominator:
\[ \tan(x + y) = \frac{(2 + 2\sqrt{5})(4 + \sqrt{5})}{(4 - \sqrt{5})(4 + \sqrt{5})} = \frac{(2 + 2\sqrt{5})(4 + \sqrt{5})}{16 - 5} = \frac{(2 + 2\sqrt{5})(4 + \sqrt{5})}{11} \]
\[ \tan(x + y) = \frac{2(4 + \sqrt{5}) + 2\sqrt{5}(4 + \sqrt{5})}{11} = \frac{8 + 2\sqrt{5} + 8\sqrt{5} + 2(\sqrt{5})^2}{11} = \frac{8 + 10\sqrt{5} + 10}{11} = \frac{18 + 10\sqrt{5}}{11} \]

Thus,
\[ K = \frac{18 + 10\sqrt{5}}{11} \]

Would you like details or have any questions?

Here are 5 more problems you might consider:
1. Solve for \(\cos(x + y)\) given \(\cos x = \frac{1}{2}\) and \(\cos y = \frac{1}{3}\).
2. Prove the identity \(\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}\).
3. Simplify \(\sin(2x)\) given \(\cos x = \frac{3}{5}\).
4. Calculate \(\sin(2x + 2y)\) using the sum-to-product identities.
5. Determine the value of \(\sin(x - y)\) if \(\sin x = \frac{3}{5}\) and \(\sin y = \frac{4}{5}\).

Tip: When working with trigonometric identities, always double-check your angle sum and difference formulas to ensure accuracy.

Discover how to calculate K = tan(x + y) given sin(x) = 1/√5 and cos(y) = 2/3. This detailed guide explores key trigonometric concepts, formulas, and theorems for students in Grades 10-12.

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