Question

Match each value with its formula for ABC

Answer 1

Hello! The questions you posted require the application of the sine law. This law states that, for every triangle with sides labelled a, b, and c, the following relationships are true:

`(sin A)/(a)= (sin B)/(b)=(sin C)/(c)`

The uppercase letters represent the angle OPPOSITE to the sides labelled with the letters' lowercase counterparts.

1. For this item, since we can see variables a and b, we will proceed to applying the relationship containing these variables. Thus, we will have the equation
`(sin A)/(a)= (sin B)/(b)`.

To arrive at the same expression given in the question, we just simply need to multiply both sides by the variable a.

`(a*sin A)/(a)= (a*sin B)/(b)`

`sinA= (a*sin B)/(b)`

ANSWER: Thus, the expression here is equal to sin A.

2. For this one, we need to use the equation containing the variables b and c.

`(sin B)/(b)= (sin C)/(c)`

To arrive at the given expression, we just need to multiply both sides by the variable b.

`(b*sin B)/(b)= (b*sin C)/(c)`

`sinB= (b*sin C)/(c)`

ANSWER: The formula in this item is equivalent to sin B.

3. Here we can see the variables a and c so we will use the equation with those variables.

`(sin A)/(a)= (sin C)/(c)`

Like the two previous items, we just multiply both sides by a variable (c for this item) to arrive at the given formula.

`(c*sin A)/(a)= (c*sin C)/(c)`

`(c*sin a)/(a)=sinC`

ANSWER: The given formula is equivalent to sin C.

4. For this one we would need the following equation:
`(sin A)/(a)= (sin C)/(c)`.

To arrive at the given formula, we first need to perform cross multiplication. After doing so, we just proceed to perform algebraic operations until we arrive at the given formula.

`sinA*c=sinC*a` (divide both sides by sin C)

`(sinA*c)/(sinC)=a`

ANSWER: The formula in this item is equivalent to the variable a.

5. For this item we perform the same process in the previous one however we just use the equation with variables a and b.

`(sin A)/(a)= (sin B)/(b)`

`sinA*b=sinB*a` (divide both sides by sin A)

`b=(sinB*a)/(sinA)`

ANSWER: The value for the formula in this item would be the variable b.

6. Lastly, for this one we would also need to follow the same processes in the two items. We just need to use the equation with the variables b and c since it is the given.

`(sin B)/(b)= (sin C)/(c)`

`sinB*c=sinC*b` (divide both sides by sin B)

`c=(sinC*b)/(sinB)`

ANSWER: The formula is equivalent to the variable c.