Finding an Angle In a Triangle when Only 2 Sides are Known?

Question

In Triangle ABC, AC = 3cm, BC = 2cm, ∠BAC = θ and ∠ABC = 2θ. Calculate the Value of θ?

I used the Sine Rule to get the ratio of Angles to 2/3, but not sure how else to proceed. Thanks.

Answer 1

First, we draw our triangle to get the visual, and label the sides and angles correctly.  Once you have done that, you realized that you can use law of sine to find the value of the angle.  This is because we have 2 pairs of angles that are opposite to their known sides.

 

You should end up with this proportion:

 

AC / sin(ABC) = BC / sin(BAC)

 

Let   x = θ

 

 

Plug in the values into this proportion.

 

3 / sin(2x) = 2 / sin(x)

 

 

Cross-multiplying, we get

 

3sin(x) = 2sin(2x)

 

 

Divide both sides of equation by 3.

 

sin(x) = (2/3)sin(2x)

 

 

Now, using the double-angle identity on the right side of equation,

 

sin(x) = (2/3) * (2sin(x)cos(x))

 

 

Divide both sides of the equation by sin(x).

 

1 = (4 / 3) cos(x)

 

 

Multiply both sides of equation by 3/4.

 

3 / 4 = cos(x)

 

 

Take the inverse cosine on both sides of equation.

 

41.41 = x

 

41.41 = θ