Question

Find the Value of Sin A rounded to the nearest hundredth, if necessary.

Thanks if you decide to help !!

Answer 1

Explanation:

In trigonometry, the law of sines is an equation that relates the length of the sides of a triangle (any type of triangle) to the sines of its angles. This is expressed mathematically as

`\displaystyle(a)/(\sin \alpha) \ = \ \displaystyle(b)/(\sin \beta) \ = \ \displaystyle(c)/(\sin \gamma)`,

where
`a`,
`b`, and
`c` are the lengths of the sides of the triangle and
`\alpha`,
`\beta`, and
`\gamma` are the opposite angles (as shown in the figure below). Likewise, the law is sometimes written as it reciprocals,

`\displaystyle(\sin \alpha)/(a) \ = \ \displaystyle(\sin \beta)/(b) \ = \ \displaystyle(\sin \gamma)/(c)`.

Therefore,

`\displaystyle(\sin A)/(BC) \ = \ \displaystyle(\sin B)/(AC) \\ \\ \\ \sin A \ = \ \displaystyle(\sin B \ * \ BC)/(AC) \\ \\ \\\sin A \ = \ \displaystyle\frac{\sin \left(90^(\circ)\right) \ * \ \sqrt{\left(10\right)^(2) \ - \ \left(√(78)\right)^(2)}}{10} \\ \\ \\ \sin A \ = \ \dsiplaystyle(√(22))/(10) \\ \\ \\ \sin A \ = \ 0.47 \ \ \ \ (\text{nearest hundredth})`

Alternatively, you can solve this question using the definition of the trigonometric function sine. For the angle
`\theta` in the figure below,

`\sin \theta \ = \ \displaystyle\frac{\text{opposite}}{\text{hypothenuse}}`.

Therefore,

`\sin A \ = \ \displaystyle(BC)/(AC) \\ \\ \\ \sin A \ = \ \displaystyle\frac{\sqrt{\left(10\right)^(2) \ - \ \left(√(78)\right)^(2)}}{10} \\ \\ \\ \sin A \ = \ \displaystyle(√(22))/(10) \\ \\ \\ \sin A \ = \ 0.47 \ \ \ \ (\text{nearest hundredth})`