Question

For each triangle shown below, determine whether you would use the Law of Sines or Law of Cosines to find angle x. Then find angle x to the nearest tenth.

Answer 1

34.9° by the Law of Cosines

Explanation:

To solve this, we first have to know when to use the Law of Sines and when to use the Law of Cosines.

When to use the Law of Sines:

Use the Law of Sines when you have two angles and a side, or two sides and an angle, and you want to find the missing angle or side.

The Law of Sine states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle:

`\boxed{(a)/(sinA) = (b)/(sinB) = (c)/(sinC)}`.

When to use the Law of Cosines:

Use the Law of Cosines when you have two sides and the included angle, or all three sides of a triangle, and you want to find the missing angle or side.

The Law of Cosines states:

`\boxed{a^2 = b^2 + c^2 - 2bc \ cos(A)}`,

where A is the angle included between sides b and c.

For the triangle given in the question, we have all three sides and need to find a missing angle. Therefore the Law of Cosines must be used:

`{a^2 = b^2 + c^2 - 2bc \ cos(A)}`

⇒ 18² = 30² +30² - 2(30)(30) × cos(
`x`)

⇒ 324 = 1800 - 1800·cos(
`x`)

⇒ 1800·cos(
`x`) = 1800 - 324

⇒ 1800·cos(
`x`) = 1476

⇒ cos(
`x`) =
`(1476)/(1800)`

⇒ cos(
`x`) = 0.82

⇒
`x` = cos⁻¹(0.82)

⇒
`x` = 34.9°