The law of cosines
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The law of cosines is a generalization of the Pythagorean theorem that can be used to find unknown sides or angles of any triangle, not just right triangles. It states that c^2 = a^2 + b^2 - 2abcos粒, where c is the side opposite angle 粒. If 粒 is a right angle, it reduces to the Pythagorean theorem. The law of cosines can be used for triangulation and was developed by early Muslim mathematicians, building on a similar theorem in Euclid's Elements from the 3rd century BC.
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The law of cosines is a generalization of the Pythagorean theorem that can be used to find unknown sides or angles of any triangle, not just right triangles. It states that c^2 = a^2 + b^2 - 2abcos粒, where c is the side opposite angle 粒. If 粒 is a right angle, it reduces to the Pythagorean theorem. The law of cosines can be used for triangulation and was developed by early Muslim mathematicians, building on a similar theorem in Euclid's Elements from the 3rd century BC.9 4 the law of cosines
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Here are the steps to solve this problem:
a. AC is the hypotenuse of the right triangle ABC.
We are given that AB = 6 units
and angle BAC = 45 degrees
Using the trigonometric identity that sin(45) = 0.5, we can set up:
sin(45) = opposite/hypotenuse
0.5 = AB/AC
AC = AB/sin(45)
AC = 6/0.5 = 12 units
b. To find BC, we can use the Pythagorean theorem since this is a right triangle:
BC^2 = AB^2 + AC^2 - 2AB*ACcos(B)
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- It can be used when you know an angle and side opposite (SAA) or have two sides and an angle (SSA) or have an angle, side, and another angle (ASA)
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The document discusses the Law of Sines, which can be used to solve triangles that are not right triangles. It provides three formulas for finding the area of a triangle using two angles and one included side: Area = 1/2 * b * c * sin(A), 1/2 * a * c * sin(B), 1/2 * a * b * sin(C). It also gives the formula for the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). An example problem finds the area of a triangle given two side lengths and one angle measure.4th l6. oblique triangle
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We are given that AB = 6 units
and angle BAC = 45 degrees
Using the trigonometric identity that sin(45) = 0.5, we can set up:
sin(45) = opposite/hypotenuse
0.5 = AB/AC
AC = AB/sin(45)
AC = 6/0.5 = 12 units
b. To find BC, we can use the Pythagorean theorem since this is a right triangle:
BC^2 = AB^2 + AC^2 - 2AB*ACcos(B)
BC00214
00214nasmaazam
油
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Trigonometry is the branch of mathematics dealing with relationships between the sides and angles of triangles. It has been developed and used for over 4000 years, originating in ancient civilizations for purposes like calculating sundials. A key foundation is the right triangle, where one angle is 90 degrees. Pythagoras' theorem relates the sides of a right triangle, and trigonometric ratios like sine, cosine, and tangent define relationships between sides and angles. Trigonometry has many applications, from astronomy and navigation to engineering, physics, and digital imaging.犖幡験犢仰犖犖犖犖犖犖朽犖犖犖÷鹸犖犖巌犖犖巌
犖幡験犢仰犖犖犖犖犖犖朽犖犖犖÷鹸犖犖巌犖犖巌krunittayamath
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Trigonometry deals with relationships between sides and angles of triangles, especially right triangles. It has been used for thousands of years in fields like astronomy, navigation, architecture, engineering, and more modern fields like digital imaging and computer graphics. Trigonometric functions define ratios between sides of a right triangle and are used to solve for unknown sides and angles. Common applications include calculating distances, heights, satellite positioning, and modeling waves and vibrations.The Axiom - Issue 17 - St Andrew's Day 2024
The Axiom - Issue 17 - St Andrew's Day 2024EtonSTEM
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The Axiom is Eton College's flagship mathematics publication.Trigonometry Exploration
Trigonometry ExplorationJanak Singh saud
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The document discusses trigonometry and provides definitions and examples of its uses. It defines trigonometry as the branch of mathematics dealing with relationships involving lengths and angles of triangles. It then gives examples of how trigonometry is used in fields like navigation, architecture, engineering, and game development. It also provides information on trigonometric functions like sine waves and their importance in fields like physics and signal processing.Trigonometry
TrigonometryJordon Dsouza
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Trigonometry is a branch of mathematics that studies relationships between side lengths and angles in triangles. The key concepts are the trigonometric functions sine, cosine, and tangent, which describe ratios of sides of a right triangle. Trigonometry has applications in fields like navigation, music, engineering, and more. It has evolved significantly from its origins in ancient Greece and India, with modern definitions extending it to all real and complex number arguments.Core sub math_att_4pythagoreantheorem
Core sub math_att_4pythagoreantheoremSatyam Gupta
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The document provides information about the Pythagorean theorem:
1) It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2) It gives examples of right triangles that satisfy the theorem, such as ones with sides of 3, 4, 5 or 5, 12, 13.
3) It includes an animated proof of the theorem showing how the area of the square on the hypotenuse equals the combined areas of the squares on the other two sides.8.5 Law of Sines; Law of Cosines 2.ppt
8.5 Law of Sines; Law of Cosines 2.pptborattheboyyyy
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This document provides examples and explanations for using the Law of Sines and Law of Cosines to solve triangles. It begins with sample problems using trigonometric ratios to find values for obtuse angles. It then explains how to use the Law of Sines when given two angles and a side or two sides and a non-included angle. Similarly, it describes using the Law of Cosines when given two sides and an included angle or three sides. Several multi-step example problems demonstrate applying these laws to find missing side lengths and angle measures in various triangles. The document emphasizes showing work and not rounding until the final step.An Introduction to Trigonometry.pdf
An Introduction to Trigonometry.pdfJessica Navarro
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This document provides an introduction to trigonometry. It begins by defining the trigonometric functions based on the unit circle. It then discusses converting between degree and radian measure, and calculating trig functions of special angles like 0, /6, /4, /3, and /2 radians. The document also covers extending these calculations to angles that are multiples of the basic angles, as well as using a calculator when needed. It introduces applications of trigonometry to right triangles, and solving simple trigonometric equations. Finally, it discusses some important trigonometric identities.Trigonometry
Trigonometryshikhar199
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Trigonometry is the branch of mathematics that deals with relationships between sides and angles of triangles, especially right triangles. It has its origins in ancient civilizations over 4000 years ago and was originally used to calculate sundials. Key concepts in trigonometry include the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) that relate ratios of sides and angles in triangles. Trigonometric functions have many applications in fields like astronomy, navigation, engineering, and more.TRIGONOMETRY
TRIGONOMETRYRoyB
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Trigonometry is the branch of mathematics dealing with relationships between the sides and angles of triangles and the calculations of trigonometric functions. It has many applications in fields like navigation, surveying, physics, engineering and more. Some key aspects covered are the definitions of trigonometric functions like sine, cosine and tangent; common trigonometric identities; formulae for calculating unknown sides and angles of triangles; and the history of trigonometry dating back to ancient Greek and Indian mathematicians.English for Math
English for MathAmalia Indrawati Gunawan
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Trigonometry is the branch of mathematics that studies relationships involving lengths and angles of triangles. Key concepts include:
- The trigonometric functions (sine, cosine, tangent, etc.) relate angles and side lengths of triangles and are most simply defined using the unit circle.
- Trigonometry has its roots in ancient Greek, Indian, Chinese, Islamic and European mathematics. Important early contributors include Hipparchus, Ptolemy and Aryabhata.
- Right-angled triangle definitions establish the hypotenuse, opposite and adjacent sides and define the trigonometric functions in terms of ratios of these sides. trigonometry and application 
trigonometry and application TRIPURARI RAI
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Trigonometry is derived from Greek words meaning "three angles" and "measure". It deals with relationships between sides and angles of triangles, especially right triangles. The document discusses the history of trigonometry dating back to ancient Egypt and Babylon, and how it advanced through the works of Greek astronomer Hipparchus and Ptolemy. It also discusses the six trigonometric ratios and their formulas, various trigonometric identities, and applications of trigonometry in fields like architecture, engineering, astronomy, music, optics, and more.More from Terry Gastauer (8)
Lesson 1 1 properties of real numbers
Lesson 1 1 properties of real numbersTerry Gastauer
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1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
2) It discusses ordering real numbers and using symbols like <, >, , to compare them. Properties of addition, multiplication and other operations are also covered.
3) Examples are provided to illustrate concepts like using properties of real numbers to evaluate expressions and convert between units like miles and kilometers.Roman numbers
Roman numbersTerry Gastauer
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Roman numerals use symbols to represent numbers, with I=1, V=5, X=10, L=50, C=100, D=500, and M=1000. Numbers are written by combining symbols, with smaller values subtracted from larger ones to the left. For example, IV=4 and IX=9. Only one digit can be subtracted at a time. Roman numerals cannot have a number subtracted that is more than 10 times larger. Practice problems are provided to convert between Hindu-Arabic and Roman numerals.Coordinate proofs
Coordinate proofsTerry Gastauer
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This document discusses using coordinate geometry to prove theorems about shapes in geometry. It states that it is easiest to work with shapes that are aligned with the x- and y-axes, like right triangles oriented with legs parallel to the axes or trapezoids and parallelograms aligned with the axes. It provides examples of using coordinates to prove properties about the midpoint of a hypotenuse, the median of a trapezoid, and the altitudes of a triangle. Finally, it lists five methods used in coordinate proofs, such as using the distance formula to prove line segments are equal in length.The law of cosines
The law of cosinesTerry Gastauer
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The law of cosines is a generalization of the Pythagorean theorem that can be used to find unknown sides or angles of any triangle, not just right triangles. It states that c^2 = a^2 + b^2 - 2abcos粒, where c is the side opposite angle 粒. If 粒 is a right angle, it reduces to the Pythagorean theorem. The law of cosines can be used for triangulation and was developed by early Muslim mathematicians, building on a similar theorem in Euclid's Elements from the 3rd century BC.The law of cosines
The law of cosinesTerry Gastauer
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The law of cosines is a generalization of the Pythagorean theorem that can be used to find unknown sides or angles of any triangle, not just right triangles. It states that c^2 = a^2 + b^2 - 2abcos粒, where c is the side opposite angle 粒. If 粒 is a right angle, it reduces to the Pythagorean theorem. The law of cosines can be used for triangulation and was developed by early Muslim mathematicians, building on a similar theorem in Euclid's Elements from the 3rd century BC.
The law of cosines
The law of cosinesTerry Gastauer
油
The law of cosines is a generalization of the Pythagorean theorem that can be used to find unknown sides or angles of any triangle, not just right triangles. It states that c^2 = a^2 + b^2 - 2abcos粒, where c is the side opposite angle 粒. If 粒 is a right angle, it reduces to the Pythagorean theorem. The law of cosines can be used for triangulation and was developed by early Muslim mathematicians, building on a similar theorem in Euclid's Elements from the 3rd century BC.Pythagorean triples
Pythagorean triplesTerry Gastauer
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Pythagorean triples are sets of three whole numbers that satisfy the Pythagorean theorem. A primitive Pythagorean triple has no common factors between the three numbers. Euclid developed a formula to generate primitive Pythagorean triples using two integers where one is odd and they are relatively prime. The document discusses using Euclid's formula to find primitive Pythagorean triples and prove the formula, as well as properties of primitive Pythagorean triples.The law of cosines
- 1. The Law of Cosines: A Generalization of the Pythagorean Theorem
- 2. The Pythagorean Theorem provides a method to find a missing side for a right triangle. But what do we do for triangles that are not right?The law of cosines states that: c2 = a2 + b2 - 2ab cos粒. Notice that if 粒 = 90属, the equation reduces to the Pythagorean Theorem since cos90属 = 0. Why do we use the Law of Cosines?Figure 1
- 3. The law of cosines can be useful in triangulation, a process for solving a triangles unknown sides and angles when only certain information about that triangle is given. For example, it can be used to find: the third side of a triangle if two sides are known and the angle between them is also known.the angles of the triangle if one knows the three sides.the third side of a triangle if two sides are known and an opposite angle to one of those sides is known. Applications
- 4. The cosine function first arose from the need to compute the sine of the complementary angle (90属 - 留). The general law of cosines was not developed until the early part of the 10th century by Muslim mathematicians. Brief History of the Law of CosinesFigure 2
- 5. An early geometric theorem which is equivalent to the law of cosines was written in the 3rd century B.C. in Euclids Elements. This theorem stated the following: Proposition 12In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle. Euclid's Elements, translation by Thomas L. Heath.Using the figure below, Euclids statement can be put into the following algebraic terms: AB2 = CA2 + CB2 +2(CA)(CH) Law of Cosines in Euclids ElementsFigure 3
- 6. There are many proofs for the Law of Cosines that use: The distance formulaTrigonometryThe Pythagorean TheoremPtolemys TheoremArea ComparisonCircle GeometryVectorsProofs for the Law of Cosines
- 7. To demonstrate your knowledge of the Law of Cosines and how it is a direct result of the Pythagorean Theorem, please answer one of the questions in the assessment section titled 3a - Assessment which can be found in the sidebar on the left of this page. Learning Activities