교과서 질문
In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i - 4j, w = -2i - j
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
4장, 문제 3
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. B = 66°, a = 17, c = 12
검증된 단계별 안내1
Identify the given elements of the triangle: angle \(B = 66^\circ\), side \(a = 17\), and side \(c = 12\). We need to find the remaining sides and angles of the triangle.
Use the Law of Sines, which states \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), to find angle \(C\) or angle \(A\). Since we know \(a\), \(c\), and \(B\), start by finding angle \(C\) using \(\frac{a}{\sin A} = \frac{c}{\sin C}\) or find \(A\) first by relating \(a\) and \(B\).
Calculate angle \(A\) using the Law of Sines: \(\frac{a}{\sin A} = \frac{c}{\sin C}\) can be rearranged to find \(\sin A\) if you find \(\sin C\) first, or alternatively use \(\frac{a}{\sin A} = \frac{b}{\sin B}\) if \(b\) is known. Since \(b\) is unknown, try to find \(\sin A\) by using \(\frac{a}{\sin A} = \frac{c}{\sin C}\) and express \(\sin C\) in terms of \(B\) and \(A\).
Check for the possibility of two triangles by considering the ambiguous case of the Law of Sines (SSA configuration). Calculate \(\sin A\) and verify if it is less than or equal to 1. If \(\sin A < 1\), then two possible angles for \(A\) exist: \(A\) and \(180^\circ - A\). This means two triangles may be possible.
Once angles \(A\) and \(C\) are found, use the fact that the sum of angles in a triangle is \(180^\circ\) to find the missing angle. Then use the Law of Sines again to find the missing side \(b\). Round all lengths to the nearest tenth and angles to the nearest degree as required.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Law of Sines
The Law of Sines relates the ratios of sides to the sines of their opposite angles in any triangle: (a/sin A) = (b/sin B) = (c/sin C). It is essential for solving triangles when given two angles and a side or two sides and a non-included angle, as in this problem.
추천 영상:
가이드 코스
4:27
Intro to Law of Sines
Ambiguous Case of the Law of Sines (SSA Condition)
When two sides and a non-included angle (SSA) are given, there can be zero, one, or two possible triangles. This ambiguity arises because the given angle and side lengths may produce no triangle, a unique triangle, or two distinct triangles, requiring careful analysis.
추천 영상:
가이드 코스
9:50Solving SSA Triangles ("Ambiguous" Case)
Triangle Angle Sum Property
The sum of the interior angles in any triangle is always 180°. After finding one unknown angle using the Law of Sines, this property helps determine the remaining angle, completing the triangle's angle measures.
추천 영상:
4:47Sum and Difference of Tangent
관련 실천
교과서 질문
Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.
cos 7x cos 3x
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교과서 질문
In oblique triangle ABC, C = 68°, a = 5, and b = 6. Find c to the nearest tenth.
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교과서 질문
In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.
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교과서 질문
In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
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교과서 질문
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. B = 107°, C = 30°, c = 126
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