Question

The legs of a right triangle are lengths x and x square root 3. The cosine of the smallest angle of the triangle is _____.

Answer 1

AC=x, BC=x√3, AB is a hypotenuse (look at the pic below)
x√3>x, consequently ∠B<∠A and we have to find cosB, which is (x√3)/AB.
Let's find AB:
AB=√(x²+(x√3)²)=√(x²+3x²)=√(4x²)=2x
Thus, cosB is:
cosB=(x√3)/(2x)=√3/2

Answer 2

The cosine of the smallest angle of the triangle is
`(√(3))/(2)`

Explanation:

see the attached figure to better understand the problem

we know that

In the right triangle ABC

Applying the Pythagoras Theorem

Find the length of the hypotenuse AB

`AB^(2)=AC^(2)+BC^(2)`

substitute the values

`AB^(2)=x^(2)+(x√(3))^(2)`

`AB^(2)=x^(2)+3x^(2)`

`AB^(2)=4x^(2)`

`AB=2x`

In the right triangle ABC

The smallest angle is the angle opposite to the smallest side

therefore

the angle B is the smallest angle

Remember that

`cos(B)=(BC)/(AB)`

substitute

`cos(B)=(x√(3))/(2x)`

`cos(B)=(√(3))/(2)`