Law of Sines in Algebra

2 Key excerpts on "Law of Sines in Algebra"

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  eBook - ePub

  Math Starters

  5- to 10-Minute Activities Aligned with the Common Core Math Standards, Grades 6-12

  • Judith A. Muschla, Gary Robert Muschla, Erin Muschla(Authors)
  • 2013(Publication Date)
  • Jossey-Bass

    (Publisher)

  .

  Natalie said that the formula for finding the area of a triangle can be used to prove the law of sines.

  She explained that the area of the three congruent triangles above can be written in three ways. Since the triangles are congruent, their areas are the same. Because of this, Natalie says that the law of sines is apparent. But her friend Emily said that it was not apparent. She said, correctly, that some steps of Natalie's reasoning are missing.

  Problem: What are the missing steps?

  5-73 Using the Law of Sines (G-SRT.11)

  The law of sines, which applies to any states that . You can always find a missing side or missing angle of a triangle if you are given one side and two angles, or two sides and the angle opposite one of the sides.

  Problem: A surveyor wants to determine the distance across a river from to as shown in the figure below.

  He knows the distance from to is 150 feet. He selected a point and found and . Find BC, rounded to the nearest foot.

  5-74 Applying the Law of Cosines (G-SRT.11)

  The law of cosines states the following for any :

  The law of cosines can be used to find the measure of a side or angle of a triangle if you know the measures of three sides or the lengths of two sides and the angle between them.

  Problem: To determine the distance across a lake, a surveyor selected a point and found and , as shown on the figure below. Find the distance across the lake, rounded to the nearest tenth of a mile.

  5-75 Identifying Types of Quadrilaterals

  A quadrilateral is any four-sided polygon. Common types of quadrilaterals are listed below:

  • A trapezoid is a quadrilateral that has only one pair of parallel sides.
  • A parallelogram is a quadrilateral whose opposite sides are parallel and congruent.
  • A rhombus is a parallelogram that has four congruent sides.
  • A rectangle is a parallelogram that has four right angles.
  • A square is a rectangle that has four congruent sides. A square is a rhombus that has four right angles.

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  eBook - ePub

  Mathematician's Delight

  • W. W. Sawyer(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  less to the East. The record appears as below :

  The first problem assumes that we have already in our possession satisfactory tables of sines and cosines, and is known as the solution of triangles. It is a problem which naturally arises in surveying. We are given certain information about a triangle, sufficient to enable us to draw the triangle, and are asked to find the remaining quantities. For instance, in any triangle ABC we might be told the length of AB, and the angles ABC and BAC, and asked to find the lengths AC and BC. This problem frequently arises in map-making, in the construction of range-finders, in determining the position of a ship at sea by taking the bearings of two lighthouses, in locating submarines, etc.

  Surveyors and seamen are in a position to buy printed books of tables, containing sines and cosines and other information. But someone first had to make these tables — this is our second problem — and several of the properties of sines and cosines were discovered with this object in view. The interest which mathematicians of the sixteenth century showed in algebra was partly due to the fact that equations had to be solved before trigonometric tables could be made.10

  Thirdly, it is desirable to know the properties of sines and cosines on quite general grounds. They arise in many problems, and the work can often be made shorter and simpler if the formulae are known. An example of this will be given later.

  Pythagoras’ Theorem

  In Fig. 17 the sides OQ and QP have the lengths cos t and sin t, where t is short for the angle QOP. Students usually find this figure easy enough to grasp, but they do not always recognize it when it occurs in an unusual position, or on a different scale. For instance, in Fig. 18 , DF makes an angle t with DE, and DF has the length 1. The line EG is drawn at right angles to DF. It is clear enough that the triangle DEF has the same shape as the triangle OQP. It may not be so obvious that there are two other triangles in the figure with the same shape. But this is so. If you cut out pieces of paper just large enough to cover the triangles DEG and FEG, you will find that it is possible (after turning the paper over) to lay these triangles in the positions LVW and LTU. The triangle LMN is exactly the same size and shape as DEF.

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