Geometry Answer Note #11

Question 1 Solve each triangle ABC that exists. Bequals34 degrees 42 prime aequals38.5 bequals31.1 Question content area bottom Part 1 Select the correct choice below and, if necessary, fill in the answer boxes within the choice. A. There is only one possible solution for the triangle. The measurements for the remaining angles A and C and side c are as follows. Aequals enter your response heredegrees enter your response hereprime Cequals enter your response heredegrees enter your response hereprime cequals enter your response here (Simplify your answer. Round to the nearest degree as needed. Round to the nearest minute as needed.) (Round to the nearest tenth as needed.) B. There are two possible solutions for the triangle. The measurements for the solution with the longer side c are as follows. Upper A 1equals enter your response heredegrees enter your response hereprime Upper C 1equals enter your response heredegrees enter your response hereprime c 1equals enter your response here (Simplify your answer. Round to the nearest degree as needed. Round to the nearest minute as needed.) (Round to the nearest tenth as needed.) The measurements for the solution with the shorter side c are as follows. Upper A 2equals enter your response heredegrees enter your response hereprime Upper C 2equals enter your response heredegrees enter your response hereprime c 2equals enter your response here (Simplify your answer. Round to the nearest degree as needed. Round to the nearest minute as needed.) (Round to the nearest tenth as needed.) C. There are no possible solutions for this triangle. Answer Based on the given information, we can start by drawing the triangle and labeling the known angles and sides as follows: A: unknown angle B: 34 degrees 42 prime C: 90 degrees a: 38.5 b: 31.1 c: unknown side To find the remaining angles and side, we can use the Law of Cosines, which states that c^2 = a^2 + b^2 - 2abCos(C). Step 1: Find c^2 c^2 = a^2 + b^2 - 2abCos(C) c^2 = (38.5)^2 + (31.1)^2 - 2(38.5)(31.1)Cos(90) c^2 = 1487.25 + 966.81 - 2366.95(0) c^2 = 2454.06 Step 2: Take the square root of both sides to find c c = √2454.06 c = 49.54 Therefore, the length of side c is 49.54. Step 3: Find angle A using the Law of Cosines again Cos(A) = (b^2 + c^2 - a^2) / 2bc Cos(A) = (31.1)^2 + (49.54)^2 - (38.5)^2 / 2(31.1)(49.54) Cos(A) = 1782.21 / 3067.58 Cos(A) = 0.5811 A = Cos^-1(0.5811) A = 54.84 degrees Step 4: Find angle C using the fact that the sum of the angles in a triangle is 180 degrees C = 180 - A - B C = 180 - 34.70 - 90 C = 55.30 degrees Therefore, the measurements for triangle ABC are: A = 54.84 degrees B = 34 degrees 42 prime C = 55.30 degrees a = 38.5 b = 31.1 c = 49.54 Question 2 if sin thita =2/3, find all other trigonometric ratios of angle theta Answer To find the other trigonometric ratios of angle theta, we will first need to understand what sine (sin) represents in trigonometry. Sine is a Trigonometric function which represents the ratio of the side opposite to given angle theta to the hypotenuse of a right triangle. It is important to note that, in a right triangle, the hypotenuse is always the longest side while the side opposite to the angle is the shortest. With this in mind, let's dive into finding the other trigonometric ratios. Cosine (cos) is another trigonometric function that is used to find the ratio of the adjacent side to the hypotenuse of a right triangle. In our given question, theta is the angle we are concerned with, and we know that sine is equal to 2/3. To find the cosine of theta, we will need to use the Pythagorean theorem which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length of the other two sides. In this case, we have the length of the hypotenuse (which is the same as the denominator of sine, i.e. 3) and the length of the opposite side (which is equal to 2). Using a^2 + b^2 = c^2 formula, we can solve for the length of the adjacent side, which will give us the cosine of theta. So, cos theta = a/c = (3^2 - 2^2)/3 = 5/3. Similarly, we can find the value of the remaining trigonometric functions, tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). Tan theta = sin theta/cos theta = (2/3)/(5/3) = 2/5. Cot theta = 1/tan theta = 1/(2/5) = 5/2. Sec theta = 1/cos theta = 1/(5/3) = 3/5. Csc theta = 1/sin theta = 1/(2/3) = 3/2. In summary, the values of all trigonometric ratios for angle theta, given that sin theta = 2/3, are: sin theta = 2/3 cos theta = 5/3 tan theta = 2/5 cot theta = 5/2 sec theta = 3/5 csc theta = 3/2.