Question

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In WXY, y=690 cm, w=440 cm and angle X=163°. Find angle W to the nearest degree

Answer 1

The angle W is approximately 7°.

Explanation:

Since angle X is adjacent to sides y and w and opposite to side x, we calculate the length of side x by Law of the Cosine:

`x = √(y^2+w^2-2\cdot y\cdot w \cdot \cos X)` (1)

Where:

`y, z` - Side lengths, in centimeters.

`W` - Angle, in sexagesimal degrees.

If we know that
`y = 690\,cm`,
`w = 440\,cm` and
`X = 163^(\circ)`, then the length of the side x is:

`x = √(y^2+w^2-2\cdot y\cdot w \cdot \cos X)`

`x\approx 1118.199\,cm`

By Geometry we know that sum of internal angles in a triangle equals 180°. If X is an obtuse, then Y and W are both acute angles. By Law of the Sine we find angle W:

`(x)/(\sin X)= (w)/(\sin W)` (2)

`\sin W = \left((w)/(x) \right)\cdot \sin X`

`W = \sin^(-1)\left[\left((w)/(x) \right)\cdot \sin X\right]`

If we know that
`X = 163^(\circ)`,
`w = 440\,cm` and
`x\approx 1118.199\,cm`, then the angle W is:

`W = \sin^(-1)\left[\left((w)/(x) \right)\cdot \sin X\right]`

`W \approx 6.606^(\circ)`

Hence, the angle W is approximately 7°.