Final answer:
To find the unknown sides and angles of a triangle with given sides a = 13 and c = 25, we can use the Law of Cosines and the Law of Sines. Using the Law of Cosines, we find that side b ≈ 16.62. Using the Law of Sines, we find that angle A ≈ 52.71 degrees. Angle B can be found by subtracting angles A and C from 180 degrees, resulting in angle B ≈ 37.29 degrees.
Step-by-step explanation:
To find the unknown sides and angles of a triangle, we can use trigonometric ratios. Let's start by finding side b using the Law of Cosines.
The Law of Cosines states that for any triangle with sides a, b, and c, a² = b² + c² - 2bc*cos(A). We are given a = 13 and c = 25. Rearranging the equation and plugging in the known values, we get b² = 13² + 25² - 2*13*25*cos(A), which simplifies to b ≈ 16.62.
To find angle A, we can use the Law of Sines. The Law of Sines states that for any triangle with angles A, B, and C, and sides a, b, and c (opposite to their respective angles), a/sin(A) = b/sin(B) = c/sin(C).
We can rearrange the equation to solve for angle A: sin(A) = (a * sin(B))/b. Plugging in the known values, sin(A) ≈ (13 * sin(B))/16.62. Using the inverse sine function on a calculator, we can find that angle A is approximately 52.71 degrees.
To find angle B, we can use the fact that the sum of the angles in a triangle is 180 degrees. Angle B = 180 - angle A - angle C. Since angle A is 52.71 degrees and angle C is the right angle (90 degrees), we can substitute those values and find that angle B ≈ 37.29 degrees.