Textbook Question
Find the area of each triangle ABC.
A = 59.80°, b = 15.00 cm, C = 53.10°
542
views
7. Non-Right Triangles
Law of Sines
Problem 7.9
Textbook Question
Determine the number of triangles ABC possible with the given parts.
c = 50, b = 61, C = 58°
Verified step by step guidance1
Step 1: Use the Law of Sines to find angle B. The Law of Sines states \( \frac{b}{\sin B} = \frac{c}{\sin C} \). Substitute the known values: \( \frac{61}{\sin B} = \frac{50}{\sin 58^\circ} \).
Step 2: Solve for \( \sin B \) by rearranging the equation: \( \sin B = \frac{61 \cdot \sin 58^\circ}{50} \).
Step 3: Determine if \( \sin B \) is a valid sine value (i.e., between -1 and 1). If \( \sin B \) is greater than 1, no triangle is possible. If \( \sin B \) is less than or equal to 1, proceed to find angle B.
Step 4: Calculate angle B using the inverse sine function: \( B = \sin^{-1}(\sin B) \).
Step 5: Use the fact that the sum of angles in a triangle is 180° to find angle A: \( A = 180^\circ - C - B \). Check if the calculated angles form a valid triangle (all angles must be positive).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Law of Sines
The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of the triangle. This law is particularly useful for solving triangles when given two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA).
Recommended video:
4:27
Intro to Law of Sines
Triangle Inequality Theorem
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem is essential for determining the possibility of forming a triangle with given side lengths. It helps in assessing whether the given parts can indeed form a valid triangle.
Recommended video:
5:19Solving Right Triangles with the Pythagorean Theorem
Ambiguous Case of SSA
The Ambiguous Case of SSA occurs when two sides and a non-included angle are known, which can lead to zero, one, or two possible triangles. This situation arises because the given angle may not uniquely determine the triangle's shape. Understanding this case is crucial for determining how many triangles can be formed with the provided measurements.
Recommended video:
9:50Solving SSA Triangles ("Ambiguous" Case)
4:27mWatch next
Master Intro to Law of Sines with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice