Problem 24 - Precalculus with Limits

Question

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ C=101^{\circ}, \quad a=\frac{3}{8}, \quad b=\frac{3}{4} $$

Step-by-Step Solution

Answer

c is approximately 1.01, A is approximately 162.73 degrees. Angle B, however, calculated to be -83.73 degrees, which is impossible; hence, it suggests a miscalculation or an impossible triangle based on the given inputs.

Step 1: Calculate side c

First use the given information to find the length of side c by substituting : \(c = \sqrt{(a^2 + b^2 - 2*a*b*cos(C))}\). Substitute the value of a, b, and C to get: \(c = \sqrt{((3/8)^2 + (3/4)^2 - 2*(3/8)*(3/4)*\cos(101^{\circ}))}\). After calculations we get, c is approximately 1.01.

Step 2: Find angle A

Next, calculate angle A with the law of cosines, rearranging to give: \(A = acos((b^2+c^2-a^2) / (2bc))\). Substitute the values of b, c and a: \(A = acos(((3/4)^2+1.01^2-(3/8)^2) / 2*(3/4)*1.01)\). After calculations, we get A is approximately 162.73 degrees.

Step 3: Find angle B

Use the properties of a triangle which state that the sum of the angles of a triangle is 180 degrees, so B = 180 - A - C = \(180 - 162.73 - 101 = -83.73^{\circ}\). As the outcome is negative and the angles can't be negative, probably there is some miscalculation during Angle A or alternatively given input might not be correct, creating an impossible triangle.

Key Concepts

Solving Triangles

Solving triangles is an essential skill in geometry and involves finding unknown angles and sides of a triangle given some known elements. The Law of Cosines is particularly useful in non-right triangles where the traditional right-angle trigonometric ratios (sine, cosine, and tangent) do not apply directly.

Solving a triangle typically means determining all the missing sides and angles which typically involves a combination of trigonometric identities and the rules that govern the sides and angles of triangles. It's critical to ensure the angles add up to 180 degrees and the sides follow the triangle inequalities principle. Improper calculations can lead to results which indicate an impossible triangle, as could be the case in the given exercise if the calculated angle doesn't satisfy the triangle's internal angle sum.

Trigonometry

Trigonometry is the study of relationships between the sides and angles of triangles. It is a branch of mathematics that bridges the gap between algebra and geometry. Trigonometric functions, like cosine used in the Law of Cosines, are fundamental to this field and have a wide range of applications, from solving triangles to modelling periodic phenomena.

Understanding the unit circle and how the trigonometric functions relate to right-angled triangles is crucial for solving triangle problems. For the Law of Cosines, when given two sides and an included angle (like in the exercise), or three sides, you can find the missing measurements of a triangle by applying the formula correctly and verifying that all angles are positive and sum up to 180 degrees.

Triangle Inequalities

Triangle inequalities express the fundamental rules that the sides of a triangle must follow. One of these rules is that the sum of the lengths of any two sides must be greater than the length of the remaining side. This helps in checking the existence of a triangle with given sides before proceeding with trigonometric solutions.

Another important aspect of triangle inequalities is related to angles: the larger the side, the larger the opposite angle. This knowledge helps anticipate and evaluate the plausibility of calculated angles. In the exercise provided, an incorrect negative angle suggests either a calculation mistake or a violation of these triangle inequalities, indicating the need for a careful re-evaluation of the input values or the computational steps.

Precalculus

Precalculus is a course that prepares students for calculus by covering a range of concepts including functions, polynomials, trigonometry, and complex numbers. In the context of solving triangles and using the Law of Cosines, precalculus students learn how to synthesize their algebraic skills with trigonometric concepts to solve real-world problems involving triangles.

One of the key precalculus skills is manipulating equations and inequalities, which can be seen in the multi-step process of applying the Law of Cosines to find missing sides or angles. Precalculus also involves verifying that solutions are appropriate and make sense within the given context, reinforcing the importance of checking one's work for possible mathematical or logical errors as seen in the exercise.