Multiple Choice
Use the Law of Sines to find the angle to the nearest tenth of a degree.
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7. Non-Right Triangles
Law of Sines
3:07 minutesProblem 7
Textbook Question
In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. C = 50°, a = 3, c = 1
Verified step by step guidance1
Identify the given elements of the triangle: angle \(C = 50^\circ\), side \(a = 3\), and side \(c = 1\). We need to solve the triangle, which means finding the remaining angles \(A\) and \(B\), and side \(b\).
Use the Law of Sines, which states: \(\frac{a}{\sin A} = \frac{c}{\sin C}\). Substitute the known values to set up the equation: \(\frac{3}{\sin A} = \frac{1}{\sin 50^\circ}\).
Solve for \(\sin A\) by cross-multiplying: \(\sin A = 3 \times \frac{\sin 50^\circ}{1} = 3 \sin 50^\circ\). Calculate \(3 \sin 50^\circ\) to check if it is less than or equal to 1, which determines if a triangle exists.
If \(\sin A\) is greater than 1, then no triangle exists. If \(\sin A\) is less than or equal to 1, find angle \(A\) by taking the inverse sine: \(A = \sin^{-1}(3 \sin 50^\circ)\). Remember that the sine function can have two possible angles in the range \(0^\circ\) to \(180^\circ\), so consider both possible values for \(A\) to check if two triangles exist.
Once you find the possible values for \(A\), calculate angle \(B\) using the triangle angle sum property: \(B = 180^\circ - A - C\). Then use the Law of Sines again to find side \(b\): \(\frac{b}{\sin B} = \frac{a}{\sin A}\). Round all lengths to the nearest tenth and angles to the nearest degree.
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Video duration:
3mKey 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 ratios of the lengths of sides of a triangle to the sines of their opposite angles. It is expressed as (a/sin A) = (b/sin B) = (c/sin C). This law is essential for solving triangles when given two sides and a non-included angle (SSA), as in this problem.
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Intro to Law of Sines
Ambiguous Case of SSA Triangles
When given two sides and a non-included angle (SSA), there can be zero, one, or two possible triangles. This is known as the ambiguous case. Determining the number of solutions involves checking the height and comparing side lengths to decide if no triangle, one triangle, or two triangles exist.
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Triangle Angle Sum Property
The sum of the interior angles of any triangle is always 180°. After finding one or two unknown angles using the Law of Sines, this property helps calculate the remaining angle(s) to complete the triangle solution.
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