The document derives the Law of Sines and Law of Cosines, which relate the angles and sides of triangles. It discusses using these laws to solve oblique triangles given certain information like two angles and a side, two sides and the angle opposite one of the sides, two sides and the angle between them, or all three sides. It also covers finding the area of a triangle given two sides and the included angle, or using Heron's Formula with all three sides. Application problems demonstrate using these concepts to solve real-world geometry problems.
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
The Law of Sines is a principle of trigonometry stating that the length of the sides of any triangle are proportional to the sines of the opposite angles.
The Law of Sines is a principle of trigonometry stating that the length of the sides of any triangle are proportional to the sines of the opposite angles.
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
Triangles
Introduction
Sum of the angles of a triangle
Types of triangles
Altitude, Median and Angle Bisector
Congruence of triangles
Sides opposite congruent angles
A traverse is a series of connected lines whose lengths and directions are to be measured and the process of surveying to find such measurements is known as traversing. In general, chains are used to measure length and compass or theodolite are used to measure the direction of traverse lines.
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LEARNING OBJECTIVES
1. Develop a comprehensive understanding of the fundamental principles and concepts that form the foundation of sustainability within corporate environments.
2. Explore the sustainability implementation model, focusing on effective measures and reporting strategies to track and communicate sustainability efforts.
3. Identify and define best practices and critical success factors essential for achieving sustainability goals within organizations.
CONTENTS
1. Introduction and Key Concepts of Sustainability
2. Principles and Practices of Sustainability
3. Measures and Reporting in Sustainability
4. Sustainability Implementation & Best Practices
To download the complete presentation, visit: https://www.oeconsulting.com.sg/training-presentations
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2. DERIVATION OF LAW OF SINES Let ABC be an oblique triangle with sides a, b, and c opposite their respective angles as shown in the figure below. If an altitude h is drawn to the base, we can write the following relationship: C b a h B A c
3. Equating the two expressions for h gives Dividing both sides of the equation by sin A sin B gives the following relationship: Similarly, if we draw an altitude from angle A to side a, we can derive the following expression: Combining these two results gives the Law of Sines, summarized as follows.
4. LAW OF SINES For any triangle ABC in which a, b and c are the lengths of the sides opposite the angles with measures A, B, and C, respectively, In words, the Law of Sines may be stated as follows: The sides of a triangle are proportional to the sines of the opposite angles.
5. Solution of oblique triangles involves four cases, namely: Two angles and one side are given. Two sides and the angle opposite one of the sides are given. Two sides and the angle between those sides are given. Three sides are given. Note: The Law of Sines is applicable for the first two cases and the Law of Cosines for the last two cases.
6. Case I: Two angles and one side are given. Example: Solve the following triangles. A = 51.30B = 48.70a = 24.5 A = 410B = 570c = 52 B = 1190C = 210b = 59 Case II: Two sides and the angle opposite one of the sides are given. (AMBIGUOUS CASE) When two sides and the angle opposite one of them are given, there may be no, one, or two solutions to the triangle. For this reason, Case II is called the ambiguous case. The following are the summary of the possible cases.
7. If A is an acute angle and a < b, there are three possibilities. C C C b sin A a b b b a=b sin A a a b sin A No solution a < b sin A Two solutions a > b sin A A A A B B c B’ One solution a = b sin A
8. 2. If A is an acute angle and a ≥ b, then there is exactly one solution. C a b A B 2. If A is an obtuse angle, then there are two possibilities. c C C a a b b B A A c No solution a ≤ b One solution a > b
9. EXAMPLE: Determine how many solutions exist. When either one or two solutions exist, solve the triangle or triangles. A = 670 a = 18 b = 20 A = 870 a = 47 b = 50 A = 320 a = 7 b = 10 A = 1130 a = 49 b = 54 A = 108.70 a = 54.3 b = 51.2
10. Application: Two forest ranger stations A and B are 48 miles apart. The bearing from A to B is N700E. A ranger in each tower spots a fire. The fire’s bearing from A and B is N330E and N140W, respectively. Find the distance from the fire to each tower. The Leaning Tower of Pisa was originally approximately 56 m high. If a surveyor trying to calculate the lean of the tower walks 72 m from the center base of the tower, the angle of elevation to the top is 400. find the lean of the tower of Pisa. On a hill inclined at an angle of 15.40 with the horizontal, stands a tower. At a point Q, 61.5 m down the hill from the foot of the tower, the angle of elevation of the top of the tower is 42.60. How tall is the tower?
11. DERIVATION OF LAW OF COSINES Let ABC be an oblique triangle with sides a, b, and c opposite their respective angles as shown in the figure below. The altitude h is drawn perpendicular to the base that divides side AB into two parts: x and x-c. Using the Pythagorean theorem for each triangle gives C b a h c-x x B A c
12. Solving each of these equations for h2 gives Equating the two expressions for h2 gives Solving the equation for b2 gives From the figure:
13. Substituting this expression for x gives one form of the Law of Cosines. Using the same method and drawing altitudes to sides CB and AC gives similar results. The Law of Cosines is summarized as follows.
14. LAW OF COSINES For any triangle ABC, where a, b, and c are the lengths of the sides opposite the angles with measure A, B and C respectively,
15. Case III: Two sides and the angle between those sides are given. Example: Solve the following triangle. a = 18.4 c = 26.3 B = 47.90 C = 1150a = 11 b = 21 A = 320b = 23 c = 47 Case IV: Three sides are given. Example: Solve each triangle ABC. a = 11 b = 14 c = 17 a = 23 b = 43 c = 31
16. The Area of a Triangle The area K of any triangle ABC is given by one of these formulas: The above formulas are used to find the area of a triangle when the measures of the two sides and the included angle are known.
17. HERON’S FORMULA: If a, b, and c are the measures of the sides of a triangle, then the area K of the triangle is given by Heron’s Formula is used to find the area of a triangle when three sides are given. EXAMPLE: Find the area of the given triangle. B = 710a = 21 c = 87 a = 31 b = 23 c = 14
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