Given triangle with sides and , for which value of does ? Choose the correct value of from the options below.
Table of contents
- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
7. Non-Right Triangles
Law of Sines
Multiple Choice
According to the Law of Sines, which triangle below correctly demonstrates that the side opposite the larger angle is the larger side?
A
A triangle with angles , , and , and all sides equal
B
A triangle with angles , , and , and corresponding opposite sides , , where
C
A triangle with angles , , and , and corresponding opposite sides , , where
D
A triangle with angles , , and , and corresponding opposite sides , , where
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Verified step by step guidance1
Recall the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
Understand that this law implies the side opposite the larger angle will be longer because the sine of a larger angle (less than 180°) is greater, making the corresponding side length larger.
Examine the given triangles and their angles: for example, a triangle with angles 40°, 60°, and 80° has sides opposite these angles labeled \(a\), \(b\), and \(c\) respectively.
Apply the Law of Sines to compare side lengths: since \(\sin 80^\circ > \sin 60^\circ > \sin 40^\circ\), it follows that \(c > b > a\) for the sides opposite these angles.
Conclude that the triangle with angles 40°, 60°, and 80° and sides \(a\), \(b\), and \(c\) where \(c > b > a\) correctly demonstrates the Law of Sines relationship between larger angles and larger opposite sides.
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