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 used to find an unknown angle in a triangle when the lengths of all three sides are known, indicating the Side-Side-Side (SSS) scenario. It allows the calculation of the angle by rearranging the formula and applying the inverse cosine function. This method is valuable in various geometric problems involving triangles.
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The law of cosines can be used to find an unknown angle in a triangle if the lengths of all three sides are known (SSS). The formula a 2 = b 2 + c 2 − 2 b c cos ( A ) can be rearranged to cos ( A ) = 2 b c b 2 + c 2 − a 2 , allowing direct calculation of the angle A when a , b , and c are known. SSS
Explanation
Understanding the Law of Cosines The law of cosines relates the sides and angles of a triangle. The formula is given by:
a 2 = b 2 + c 2 − 2 b c cos ( A ) ,
where a , b , and c are the lengths of the sides of the triangle, and A is the angle opposite side a .
Rearranging the Formula To find an unknown angle measure using the law of cosines, we need to have enough information to solve for the cosine of the angle. The law of cosines can be rearranged to solve for cos ( A ) :
cos ( A ) = 2 b c b 2 + c 2 − a 2
Determining Required Information From the rearranged formula, we can see that to find the angle A , we need to know the lengths of all three sides a , b , and c . If we know the lengths of the three sides, we can directly substitute these values into the formula and calculate cos ( A ) . Then, we can find the angle A by taking the inverse cosine (arccos) of the result.
Conclusion Therefore, the law of cosines can be applied once to find an unknown angle measure on a triangle when we know the lengths of all three sides. This is known as the Side-Side-Side (SSS) case.
Examples
Imagine you're building a triangular garden and you know the lengths of all three sides. You can use the law of cosines to calculate the angles of the garden, which helps you plan the layout and ensure it fits perfectly in your yard. For example, if the sides are 5m, 7m, and 8m, you can find the angle opposite the 8m side using the law of cosines.
