Question

Solve triangle. $$a=9, c=12, \beta=23^{\circ}$$

   Solve triangle.
$$a=9, c=12, \beta=23^{\circ}$$

Precalculus

Cynthia Young 2nd Edition

Chapter 4, Problem 7

Instant Answer

Solved by Expert Sarah Lewites

06/13/2020

Step 1

We need to find side b and angles $\alpha$ and $\gamma$. We start by using the law of cosines to find side b. The law of cosines is given by $b^2 = a^2 + c^2 - 2ac\cos(\beta)$. Substituting the given values, we get: \[b^2 = 9^2 + 12^2 - Show more…

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Solve triangle. $$a=9, c=12, \beta=23^{\circ}$$

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Key Concepts

Triangle Angle Sum Property

The Triangle Angle Sum Property states that the sum of the interior angles of any triangle is always 180°. This fundamental concept is crucial when solving triangles because once two angles are known, the third can be easily found, ensuring that the complete set of angles is determined accurately.

Law of Sines

The Law of Sines connects the ratios of the lengths of sides of a triangle to the sines of their opposite angles. Once the third side is determined using the Law of Cosines in a SAS triangle, the Law of Sines can be utilized to find the other unknown angles, providing a convenient route to complete the solution of the triangle.

Law of Cosines

The Law of Cosines is used to find the unknown side of a triangle when two sides and their included angle are known. It relates the lengths of the sides of a triangle to the cosine of one of its angles, allowing one to compute the third side by taking into account the relative measure of the included angle, which is essential in solving SAS triangles.

SAS Triangle

A triangle defined by two sides and the included angle (SAS) is uniquely determined. When two sides and the angle between them are known, the triangle’s remaining side and angles can be calculated using trigonometric methods. This configuration avoids ambiguity, meaning there is only one triangle that satisfies these conditions.

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