On which triangle can the law of cosines be applied once to find an unknown angle measure? Law of cosines: [tex]a^2=b^2+c^2-2 b c \cos (A)[/tex]
Answer (2)
The law of cosines can be applied to find an unknown angle measure in a triangle if you know the lengths of all three sides. By rearranging the formula, the angle can be calculated using the inverse cosine function. This is useful in many practical scenarios where angles need to be determined from given side lengths.
;
The law of cosines relates the sides and angles of a triangle: a 2 = b 2 + c 2 − 2 b c cos ( A ) .
Rearranging the formula, we get A = arccos ( 2 b c b 2 + c 2 − a 2 ) .
To find an unknown angle using the law of cosines, we need to know the lengths of all three sides.
Therefore, the law of cosines can be applied once to find an unknown angle measure if all three side lengths are known.
Explanation
Understanding the Law of Cosines The law of cosines relates the side lengths of a triangle to the cosine of one of its angles. The formula is given by:
a 2 = b 2 + c 2 − 2 b c cos ( A )
where a , b , and c are the side lengths of the triangle, and A is the angle opposite side a . The question asks under what conditions we can apply the law of cosines to find an unknown angle measure.
Solving for the Angle We can rearrange the law of cosines to solve for the angle A :
a 2 = b 2 + c 2 − 2 b c cos ( A )
2 b c cos ( A ) = b 2 + c 2 − a 2
cos ( A ) = 2 b c b 2 + c 2 − a 2
A = arccos ( 2 b c b 2 + c 2 − a 2 )
From this formula, we can see that if we know the lengths of all three sides ( a , b , and c ), we can directly calculate the angle A using the inverse cosine function.
Other Scenarios If we know two sides (e.g., b and c ) and the included angle A , we can find the third side a using the original law of cosines: a 2 = b 2 + c 2 − 2 b c cos ( A ) . However, this only gives us the length of the third side, not an unknown angle directly.
If we know two angles and one side, we can find the third angle using the fact that the sum of the angles in a triangle is 180 degrees. Then we can use the law of sines to find the other two sides. However, this does not directly use the law of cosines to find an unknown angle measure.
Conclusion Therefore, the law of cosines can be applied once to find an unknown angle measure if all three side lengths are known.
Examples
Imagine you are building a triangular garden and you know the lengths of all three sides. Using the law of cosines, you can calculate the angles of the garden to ensure it fits perfectly in your yard. This is a practical application of the law of cosines in real-world scenarios.
