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Geometry Name______________________ Trigonometric Functions & Areas of Triangle & Law of Sines (Part II) Nov. 11 Area of a Triangle A In the past, you probably learned that the area of a triangle is given by: 1 bh 2 Question: What important relationship HAS to be true about the base and height to calculate area? It has to be true that every triangle has an area (obviously)! What may not be so obvious is how to calculate the base and height when these measures are not given about a triangle… F Consider the following triangle: e We will now generate a general formula that will enable us to determine the area of a triangle: Although every triangle has three bases, for the sake of consistency, let’s consider f as the base of the triangle for this exercise… D d f E Draw in the height (to the base f); let’s call it h. So, the formula for the area of DEF (in terms of base and height) is: Suppose the following information is given about DEF : e, (create your own tick marks for these parts) Write a mathematical equation for h, using e and (hint: use a trig function in your equation) A 1 ( f )(h) 2 D h _____________ D: Now, rewrite the formula for the area of DEF (in terms of f, e, D ): A ____________ This is a general formula for the area of any triangle! This means that any two sides and the included angle suffice to determine the area of a triangle. Given the triangle above, write the three distinct formulas for the area of the triangle using any two sides and the included angle (you can fill in the first one from your work above!): A ____________ A ____________ A ____________ Geometry Law of Sines & Its Derivation While we know that SSS, SAS, ASA, & AAS all are manners by which we can prove the uniqueness of a triangle, unfortunately these theorems/postulates do not give us an explicit way to determine the measures of the missing sides and/or angles. Fortunately, there are ways of finding the missing pieces…for now we will only consider the scenarios where we know two angles and a side, ASA and AAS. F The method we will use to find the missing pieces is called the Law of Sines! e d Let’s derive the “formula” for the Law of Sines… Below, express the area of the triangle in the three distinct ways you did earlier:(using two sides and the included angle) A ____________ D A ____________ f A ____________ Since all the above equations all provide the area of the same triangle, then all the equations can be set equal to each other… Set all the above equations equal to each other: = = Now, divide all three sides of the equation by the expression, ( 12 def ): = = Simplify the above equations and rewrite the simplified version: = = This general formula is the Law of Sines! The Law of Sines allows us to solve for unknown parts of triangles…but this only guarantees a solution to solving a triangles when given ASA or AAS. E Geometry Exercise: Draw a diagram to get started… 1) Solve the following triangle (the missing sides and angles): A 23 , B 110 , and c 50 . Sketch a reasonable diagram for this triangle. 2) Given that the area of a triangle is A 12 ab sin , where is the included angle between any two sides a and b in a triangle, calculate the area of the shaded region below. 61 12 9 8 42 10