SOLUTION: 5 cos @ + 12 sin @ = 13 Find. tan @ = ?

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Question 1062997: 5 cos @ + 12 sin @ = 13
Find. tan @ = ?
Answer by ikleyn(52797) About Me  (Show Source):
You can put this solution on YOUR website!
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5 cos @ + 12 sin @ = 13
Find tan @ = ?
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Answer. tan%28alpha%29 = 12%2F5.

Solution 

Let x = cos%28alpha%29.  Then sin%28alpha%29 = sqrt%281-cos%5E2%28alpha%29%29 = sqrt%281-x%5E2%29.


Then the original equation is equivalent to

5x+%2B+12%2Asqrt%281-x%5E2%29 = 13,   or

5x - 13 = -12%2Asqrt%281-x%5E2%29.   (1)

To solve (1), square both sides. You will get

%285x+-+13%29%5E2 = 144*(1-x^2),

25x^2 - 130x + 169 = 144 - 144x^2,

25x^2 + 144x^2 - 130x + 169 - 144 = 0,

169x^2 - 130x + 25 = 0,

%2813x+-+5%29%5E2 = 0,

---->  13x - 5 = 0  ---->  13x = 5  ---->  x = 5%2F12.


Thus you get cos%28alpha%29 = 5%2F13.


Then sin%28alpha%29 = sqrt%281-%285%2F13%29%5E2%29 = sqrt%281-25%2F169%29 = sqrt%28%28169-25%29%2F169%29 = sqrt%28144%2F169%29 = 12%2F13.


Now  tan%28alpha%29 = sin%28alpha%29%2Fcos%28alpha%29 = %28%2812%2F13%29%29%2F%28%285%2F13%29%29 = 12%2F5.