8.2 area of triangles
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The Law of Sines enables solving problems involving oblique triangles, which are triangles without right angles. It can be used when one side and any two angles of a triangle are given. The Law of Sines states that in any triangle, the ratio of any side to the sine of its opposite angle is equal to the ratios of the other two sides to their opposite angles.
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Trig 2 notes
The document is a worksheet on applying the Law of Sines to solve for unknown sides and angles of triangles. It contains examples of using the Law of Sines when two angles and a side are known, when two sides and the angle between them are known, and formulas for finding the area of a triangle given two sides and the angle between them. It includes practice problems for students to solve using the Law of Sines to find missing side lengths, angles, and areas of various triangles.
Alg2 lesson 13-5
This document discusses solving an ambiguous triangle where it is not a right triangle and the measures of angles and sides are not fully known. It determines that trigonometric functions and the Law of Sines cannot be used. It identifies that the Law of Cosines is the appropriate theorem to use to begin solving since it requires only knowing the measures of three sides.
Trig 2 notes
The document discusses the Law of Sines and how to use it to solve for missing parts of triangles. It provides two cases where the Law of Sines can be used: 1) when two angles and one side are known, and 2) when two sides and the angle between them are known. It also discusses how to find the area of a triangle using two side lengths and the angle between them. Several multi-step word problems are provided as examples of how to apply the Law of Sines to find missing side lengths and area of triangles.
Law of tangent
The document discusses the law of tangents, which sets up a ratio relationship between the sum and difference of two sides of a triangle and the tangents of half the sum and differences of the angles opposite the sides. An example problem demonstrates using the law of tangents to solve for a missing side of a triangle given two sides and one angle measure. The full working of the problem is shown, applying the tangent law formulas and then using the law of sines to find the final missing side measurement.
Law of Sines notes
The document discusses the Law of Sines, which can be used to find missing parts of any triangle. It provides the Law of Sines formula that relates the ratios of sides to opposite angles. It gives two cases where the Law of Sines can be used: 1) when two angles and any side are known, and 2) when two sides and the angle between them are known. It also provides the formula to find the area of a triangle using two sides and the included angle. Several example problems are worked through applying the Law of Sines to find missing side lengths and triangle areas.
Obj. 40 Trigonometry
For any right triangle
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
Law of sines
The document discusses the Law of Sines, which is a rule used to find unknown angles and sides of triangles when some combination of angles and sides are known. The Law of Sines states that the ratio of any side to its opposite angle is equal to the ratio of any other side to its opposite angle. An example problem demonstrates using the Law of Sines to solve a triangle when two angles and one side are given. Additional resources are provided to learn more about solving triangles with the Law of Sines.
The Sine Law (Animated)
The document introduces the Sine Law, which states that for any triangle, the ratio of the sine of an angle to the side opposite it is equal to the ratio of the sine of any other angle to its opposite side. It demonstrates this law by considering a triangle ABC and its altitude to side BC, showing that the ratio of the sine of angle A to side a equals the ratio of the sine of angle B to side b. It then states the Sine Law formula that the ratio of the sine of any angle to its opposite side equals the same ratio for any other angle and side in the triangle.
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Trig 2 notes
The document is a worksheet on applying the Law of Sines to solve for unknown sides and angles of triangles. It contains examples of using the Law of Sines when two angles and a side are known, when two sides and the angle between them are known, and formulas for finding the area of a triangle given two sides and the angle between them. It includes practice problems for students to solve using the Law of Sines to find missing side lengths, angles, and areas of various triangles.
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This document discusses solving an ambiguous triangle where it is not a right triangle and the measures of angles and sides are not fully known. It determines that trigonometric functions and the Law of Sines cannot be used. It identifies that the Law of Cosines is the appropriate theorem to use to begin solving since it requires only knowing the measures of three sides.
Trig 2 notes
The document discusses the Law of Sines and how to use it to solve for missing parts of triangles. It provides two cases where the Law of Sines can be used: 1) when two angles and one side are known, and 2) when two sides and the angle between them are known. It also discusses how to find the area of a triangle using two side lengths and the angle between them. Several multi-step word problems are provided as examples of how to apply the Law of Sines to find missing side lengths and area of triangles.
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The document discusses the law of tangents, which sets up a ratio relationship between the sum and difference of two sides of a triangle and the tangents of half the sum and differences of the angles opposite the sides. An example problem demonstrates using the law of tangents to solve for a missing side of a triangle given two sides and one angle measure. The full working of the problem is shown, applying the tangent law formulas and then using the law of sines to find the final missing side measurement.
Law of Sines notes
The document discusses the Law of Sines, which can be used to find missing parts of any triangle. It provides the Law of Sines formula that relates the ratios of sides to opposite angles. It gives two cases where the Law of Sines can be used: 1) when two angles and any side are known, and 2) when two sides and the angle between them are known. It also provides the formula to find the area of a triangle using two sides and the included angle. Several example problems are worked through applying the Law of Sines to find missing side lengths and triangle areas.
Obj. 40 Trigonometry
For any right triangle
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
Law of sines
The document discusses the Law of Sines, which is a rule used to find unknown angles and sides of triangles when some combination of angles and sides are known. The Law of Sines states that the ratio of any side to its opposite angle is equal to the ratio of any other side to its opposite angle. An example problem demonstrates using the Law of Sines to solve a triangle when two angles and one side are given. Additional resources are provided to learn more about solving triangles with the Law of Sines.
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The document introduces the Sine Law, which states that for any triangle, the ratio of the sine of an angle to the side opposite it is equal to the ratio of the sine of any other angle to its opposite side. It demonstrates this law by considering a triangle ABC and its altitude to side BC, showing that the ratio of the sine of angle A to side a equals the ratio of the sine of angle B to side b. It then states the Sine Law formula that the ratio of the sine of any angle to its opposite side equals the same ratio for any other angle and side in the triangle.
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The document discusses the Law of Sines, which can be used to solve triangles when at least one side and two other pieces of information (sides or angles) are known. It provides examples of using the Law of Sines to solve for missing sides and angles in different triangle configurations (ASA, SSA, SAA). The key points are:
- The Law of Sines states ratios of sines of angles to sides opposite are equal
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- With SSA configuration there may be one solution, no solution,
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The document provides guidance on using trigonometric rules to solve problems involving sides and angles of triangles. It explains that the Sine Rule is used when there is a "matching pair" of a known side and its corresponding angle. The Cosine Rule is used when there is no right angle or matching pair. Examples are given of using both rules to calculate missing sides or angles of triangles.
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This document discusses different methods for solving oblique triangles:
1) Case I involves being given two angles and a side opposite one of the angles.
2) Case II involves being given two angles and the included side between them.
3) Case III involves being given two sides and an angle opposite one of the sides.
4) The document provides examples of solving oblique triangles using Cases I and II.
5) Students are assigned exercises from the textbook to practice these triangle solving methods.
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Trigonometry deals with relationships between angles and sides of triangles. The three main trigonometric ratios are sine, cosine, and tangent, which relate the lengths of opposite, adjacent, and hypotenuse sides to a given angle. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Trigonometric functions like sine, cosine, and tangent can be used to calculate unknown side lengths or angles in a right triangle if one side length and its corresponding angle are known. Memorizing the definitions of the trigonometric ratios is often the most challenging aspect of learning trigonometry.
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This document discusses trigonometric ratios and their use in solving problems involving right triangles. It defines the three basic trigonometric functions - sine, cosine, and tangent - in terms of ratios of the sides of a right triangle with a given angle. Examples are provided to demonstrate calculating trigonometric ratios and using them to find missing side lengths or angle measures. The document also distinguishes between angles of elevation, which involve looking upward, and angles of depression, which involve looking downward. Sample word problems applying these angle concepts are presented at the end.
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This document introduces trigonometric ratios, which compare the sides of a right triangle. It defines the three primary ratios: sine, cosine, and tangent. Sine is the ratio of the opposite side to the hypotenuse. Cosine is the ratio of the adjacent side to the hypotenuse. Tangent is the ratio of the opposite side to the adjacent side. Several examples are provided to demonstrate calculating trigonometric ratios given a right triangle and to find angles or sides using inverse trigonometric functions.
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The document discusses the Law of Sines, which can be used to solve triangles when at least one side and two other pieces of information (sides or angles) are known. It provides examples of using the Law of Sines to solve for missing sides and angles in different triangle configurations (ASA, SSA, SAA). The key points are:
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The document provides guidance on using trigonometric rules to solve problems involving sides and angles of triangles. It explains that the Sine Rule is used when there is a "matching pair" of a known side and its corresponding angle. The Cosine Rule is used when there is no right angle or matching pair. Examples are given of using both rules to calculate missing sides or angles of triangles.
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This document discusses different methods for solving oblique triangles:
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