Answer:
[tex]x = 5[/tex]
[tex]y = \frac{5}{2}\sqrt3[/tex]
Step-by-step explanation:
Required
Find x and y
From the triangle, we can see that x is the longest side (i.e. the hypotenuse)
The sin of an angle is:
[tex]\sin(\theta) = \frac{opposite}{hypotenuse}[/tex]
The relationship between the given angle (30 degrees), x and 5/2 is:
[tex]\sin(30) = \frac{5/2}{x}[/tex]
Cross Multiply:
[tex]x * \sin(30) = \frac{5/2}{x} * x[/tex]
[tex]x * \sin(30) = \frac{5}{2}[/tex]
Solve for x
[tex]x = \frac{5}{2\sin(30)}[/tex]
[tex]\sin(30) = 0.5[/tex]
So, the expression becomes
[tex]x = \frac{5}{2*0.5}[/tex]
[tex]x = \frac{5}{1}[/tex]
[tex]x = 5[/tex]
To solve for y, we make use of Pythagoras theorem:
[tex]x^2 = y^2 + \frac{5}{2}^2[/tex]
Substitute 5 for x
[tex]5^2 = y^2 + \frac{5}{2}[/tex]
[tex]25 = y^2 + \frac{25}{4}[/tex]
Solve for [tex]y^2[/tex]
[tex]y^2 = 25 - \frac{25}{4}[/tex]
[tex]y^2 = \frac{100 - 25}{4}[/tex]
[tex]y^2 = \frac{75}{4}[/tex]
Square root of both sides
[tex]y = \sqrt{\frac{75}{4}}[/tex]
Express 75 as 25 * 3
[tex]y = \sqrt{\frac{25 * 3}{4}}[/tex]
Split:
[tex]y = \sqrt{\frac{25}{4}} * \sqrt3[/tex]
[tex]y = \frac{5}{2} * \sqrt3[/tex]
[tex]y = \frac{5}{2}\sqrt3[/tex]
