Solve for x. Round to the nearest tenth, if necessary. S 27 18° U T х
The Correct Answer and Explanation is :
To solve for ( x ) in the given triangle with sides 27 and 18° and an unknown side ( x ), we can utilize trigonometric principles. iven that one side and an adjacent angle are known, the Law of Cosines is appropriate for this scenario.
Understanding the Law of Cosines:
he Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.t is expressed as:
[ c^2 = a^2 + b^2 – 2ab \cdot \cos(C) ]
Where:
- ( c ) is the side opposite angle ( C ),- ( a ) and ( b ) are the other two sides,- ( C ) is the included angle between sides ( a ) and ( b ).
Applying the Law of Cosines:
n our case, we have:
- Side ( a = 27 )- Angle ( C = 18^\circ )- Side ( b = x ) (the unknown side)
e can rearrange the Law of Cosines to solve for ( x ):
[ x^2 = 27^2 + x^2 – 2 \times 27 \times x \times \cos(18^\circ) ]
implifying the equation:
[ x^2 = 729 + x^2 – 2 \times 27 \times x \times \cos(18^\circ) ]
ubtracting ( x^2 ) from both sides:
[ 0 = 729 – 2 \times 27 \times x \times \cos(18^\circ) ]
olving for ( x ):
[ x = \frac{729}{2 \times 27 \times \cos(18^\circ)} ]
alculating the cosine of 18°:
[ \cos(18^\circ) \approx 0.95106 ]
ubstituting this value:
[ x = \frac{729}{2 \times 27 \times 0.95106} ]
[ x \approx \frac{729}{51.45156} ]
[ x \approx 14.2 ]
Conclusion:
he length of side ( x ) is approximately 14.2 units.
his calculation demonstrates how the Law of Cosines can be applied to determine an unknown side in a triangle when two sides and the included angle are known.
or further practice and to explore more complex triangle problems, you might find the Triangle Calculator tool helpful.citeturn0search0