Find the third side of a triangle in which a 4.00-inch side and a 6.00-inch side are known to make a 56.0-degree angle. Round your answer to three significant digits.
cosine law
x^2 = 4^2 + 6^2 - 2(4)(6)cos56°
Well, let's solve this triangle puzzle. We have a 4.00-inch side, a 6.00-inch side, and a 56.0-degree angle. Now, can I be honest with you? This triangle is keeping me on the edge of my seat! *rimshot*
To find the missing side, we can use the Law of Cosines. Are you ready for some math magic? Here we go:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the missing side, a is the 4.00-inch side, b is the 6.00-inch side, and C is the 56.0-degree angle. Let's plug those values in:
c^2 = 4.00^2 + 6.00^2 - 2 * 4.00 * 6.00 * cos(56.0)
Now put on your thinking cap and calculate:
c^2 = 16.00 + 36.00 - 48.00 * cos(56.0)
c^2 = 52.00 - 48.00 * cos(56.0)
c^2 ≈ 52.00 - 24.70
Okay, let's not keep you in suspense any longer. Drumroll, please! *drumroll*
c ≈ √27.30
Rounding to three significant digits, we have:
c ≈ 5.23 inches
So, the missing side of the triangle is approximately 5.23 inches. Ta-da!
To find the third side of the triangle, we can use the Law of Cosines.
The Law of Cosines states that, for any triangle with sides a, b, and c, and angle C opposite to side c:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we know that side a is 4.00 inches, side b is 6.00 inches, and angle C is 56.0 degrees.
Let's substitute these values into the formula:
c^2 = 4.00^2 + 6.00^2 - 2 * 4.00 * 6.00 * cos(56.0)
Now, let's calculate the right side of the equation:
c^2 = 16.00 + 36.00 - 48.00 * cos(56.0)
Next, let's calculate cos(56.0) using a calculator:
cos(56.0) ≈ 0.55919 (rounded to five decimal places)
Now, substitute this value into the equation:
c^2 = 16.00 + 36.00 - 48.00 * 0.55919
c^2 = 16.00 + 36.00 - 26.7312
c^2 = 25.2688
To find the value of c, we take the square root of both sides:
c ≈ √25.2688
c ≈ 5.027 (rounded to three significant digits)
Therefore, the third side of the triangle is approximately 5.027 inches.
To find the third side of a triangle, you can use the Law of Cosines, which states that in a triangle with sides A, B, and C, and angle opposite side C, the following equation holds:
C^2 = A^2 + B^2 - 2AB * cos(C)
In this case, we know that side A is 4.00 inches, side B is 6.00 inches, and angle C is 56.0 degrees. To find side C, we can substitute these values into the equation and solve for C.
C^2 = 4.00^2 + 6.00^2 - 2 * 4.00 * 6.00 * cos(56.0)
Next, calculate the value inside the parentheses:
cos(56.0) ≈ 0.55919
Substitute this value into the equation:
C^2 ≈ 4.00^2 + 6.00^2 - 2 * 4.00 * 6.00 * 0.55919
Simplify the equation further:
C^2 ≈ 16.00 + 36.00 - 47.5112
C^2 ≈ 4.4888
Now, take the square root of both sides to solve for C:
C ≈ sqrt(4.4888)
C ≈ 2.12 inches
So, the third side of the triangle is approximately 2.12 inches when rounded to three significant digits.