Find the value of cos(alpha).

Question

given the right triangle ABC with hypotenuse c, if a=logx, b=2√logx, c=2√3, find the calue of cosα

Answer 1

I am going to assume that α is the angle opposite from side a. Let's find x first.

 

We use the pythagorean identity to get 

 

(log x)²+(2√(log x))²=(2√3)^{^2}

 

Simplifying both sides a bit, we have

 

(log x)² +4log x=12, since (2√3⁾²=4*3=12,

 

Now, we will do a substitution to make things easier. Let y = log x. This turns the equation into

 

y²+4y=12. (I just replaced every log x with a y)

 

Then, we get everything on the same side of the equation so we can factor:

 

y²+4y-12=0.

 

This factors to

 

(y+6)(y-2)=0,

 

which has solutions y=-6 and y=2.

 

Since y is the length of a side of a triangle, a negative value does not make sense, and the solution must be y=2.

 

Since y=log x, this means

 

log x=2

x=10² = 100.

 

The lengths of the two legs of the triangle are a = log x = log 100 = 2, and

b = 2√log x = 2√log 100 = 2√2.

 

Assuming α is the angle opposite a, then it is adjacent to b. The cosine is defined as the length of the adjacent leg divided by the length of the hypotenuse, so

cos α = b/c = (2√2)/(2√3)=(√2)/(√3)=(√6)/3

 

I hope this helps.