Question

Triangle.

A,
angle CAB = 90°, angle ABC = 52° and AC = 9.2.
Calculate the length of BC rounded to 3 SF.

Answer 1

`\\ \rm\Rrightarrow sin\theta=(Perpendicular)/(Hypotenuse)`

`\\ \rm\Rrightarrow sin52=(AC)/(BC)`

`\\ \rm\Rrightarrow sin52=(9.2)/(B C)`

`\\ \rm\Rrightarrow B C=(9.2)/(sin52)`

`\\ \rm\Rrightarrow B C=9.3`

Answer 2

11.7 (3 s.f.)

Explanation:

Given information:

• ∠CAB = 90°
• ∠ABC = 52°
• AC = 9.2

As one of the given angles is 90°, the triangle is a right triangle.

Draw the triangle using the given information (see attached) to help visualize the problem.

To calculate the length of BC, use the sine trigonometric ratio:

`\sf \sin(\theta)=(O)/(H)`

where:

• 
  `\theta` is the angle
• O is the side opposite the angle
• H is the hypotenuse (the side opposite the right angle)

From inspection of the attached triangle:

• 
  `\theta` = 52°
• O = AC = 9.2
• H = BC

Substitute the values into the formula and solve for BC:

`\implies \sf \sin(52^(\circ))=(9.2)/(BC)`

`\implies \sf BC=(9.2)/(\sin(52^(\circ)))`

`\implies \sf BC=11.67496758...`

`\implies \sf BC=11.7\:\:(3 \:s.f.)`

Therefore, the length of BC is 11.7 (3 s.f.).