Question

In AMNO, the measure of 0=90°, ON = 15, MO = 8, and NM = 17. What is the value of the cosine of M to the nearest hundredth?

Answer 1

To find the value of the cosine of angle M in triangle AMNO, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides
`\displaystyle a`,
`\displaystyle b`, and
`\displaystyle c`, and angle
`\displaystyle C` opposite side
`\displaystyle c`, the following equation holds:

`\displaystyle c^(2) =a^(2) +b^(2) -2ab\cos( C)`

In triangle AMNO, we have the following information:

`\displaystyle AM=17` (side
`\displaystyle a`)

`\displaystyle MN=15` (side
`\displaystyle b`)

`\displaystyle AN=8` (side
`\displaystyle c`)

Angle M = 90 degrees

We can apply the Law of Cosines to find the value of
`\displaystyle \cos( M)`:

`\displaystyle AN^(2) =AM^(2) +MN^(2) -2\cdot AM\cdot MN\cdot \cos( M)`

Substituting the given values:

`\displaystyle 8^(2) =17^(2) +15^(2) -2\cdot 17\cdot 15\cdot \cos( M)`

Simplifying:

`\displaystyle 64=289+225-510\cdot \cos( M)`

`\displaystyle 64=514-510\cdot \cos( M)`

Rearranging the equation:

`\displaystyle 510\cdot \cos( M) =514-64`

`\displaystyle 510\cdot \cos( M) =450`

Dividing both sides by 510:

`\displaystyle \cos( M) =(450)/(510)`

Simplifying:

`\displaystyle \cos( M) =(15)/(17)`

Therefore, the value of the cosine of angle M in triangle AMNO, to the nearest hundredth, is approximately
`\displaystyle 0.88`.

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