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Trigonometry: The mathematical properties of sides and angles in triangles
Trigonometry:
The mathematical properties of sides and angles in triangles
Solving for Area: There are three ways to solve for the area; Herons formula, Base and Height, and the Trigonometric Formula.
Trigonometric Formula: The Trigonometric Formula is used to find the are of a triangle when given two adjacent side to one angle. the equation goes as follows; 1/2(a)(b)(sin(ϴ)
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Step one would be to plug in the givens into the Trigonometric Formula, which are side a (30), side b (26) and Sinθ(125).
Step two would be to multiply side a and side b together. Step three would be to multiply the sum of step two by 1/2. The final step would be to multiply the sum of step three by Sin(125) to give the area units squared of the triangle.
Herons Formula: Heron's formula uses all three side of a triangle and its semiperimeter to calculate the area of the triangle.
Semiperimeter: The semiperimeter can be found by adding up all of the sides of a triangle and diving it by two. Ex. (a+b+c)/2
The equation to the herons formula is; √ (s(s-a)(s-b)(s-c)
Example
The first step to solve this is to plug in your givens into Herons Formula, which would be the semiperimeter and all three sides. Step two would be to solve everything inside the parenthesis. Step three would be to multiply everything inside the square root with the un parentheses semiperimeter. The final step is to square root the sum, which will give you the area unit squared.
Base x Height; This equation is used to find the are of a triangle based on the given height. The equation is 1/2 x base x height.
Right Triangle Formulas
Solving for Sides
Pythagorean Theorem: This theorem is used to find the missing side of a right triangle with two given sides. the equation is; a2+b2=c2. The following problem shows this equation in action.
The side gives are the side and side c. You will have to plug in what is given to find side b. This can be done by working backward. squaring side a and c, then subtracting side a from c will give you the squared length of side c. This problem will look like 12^2+b2=16^2. When it is squared it will look like 144+b=256. 144 subtracted from 256 will give you 112. b2=112 will have to be square rooted to get the final answer, which is 10.58. √b2=√112, b=10.58.
Trigonometric Ratios: With a given side and angle there are six possibilities for a given missing side to be solved.
Explanation: By identifying the triangle to be a SOH CAH TOA , its correct trigonometric equation can be used and solved for.
Ex.
For problem #4, you will need to use cosine to solve for the missing side. The given side is adjacent to the hypotenuse due to the angles position. The equation will goo as followed; Cos(36)=(8/c). The next step from this will be to isolate c. To do so, you will need to divide 8 to both sides of the equation in order to isolate c. 8/Cos(36)=8/c/8. The third step would be to successfully divide 8/Cos(36) to give you the side length which is; 9.9.
Solving for Angle: Using the SOH CAH TOA definitions, and having tow side, it is possible to find the angle.
The SOH CAH TOA ratios can be defined based on the placement of the missing angle.
Angle theta can be determined based on the Cosine ratio. This is can be determined by the sides given based on the placement of angle theta. The sides given are the hypotenuse side and the adjacent side. Plugging in the ratio into Cosines functions will give you the measure for angle theta.
The missing angle can be determined from the placement of the sides to the theta. The sides given are opposite and hypotenuse, which can be put into a ratio and plugged into the Sines function. This will give angle theta its missing measurements.
Angle theta can be determined based on Tangent ratio. With the opposite and adjacent side to angle theta determined it can be put into the Tangent function to determine the angle measurement.
Trigonometric Ratios
Identifying a Trigonometric Ratio is key to solving the measurements for a triangle. Finding the Cosine. Sines, or Tangent ratios is directly related to the angle it is based off. In a right triangle there are three side; the hypotenuse, adjacent and opposite side.
In the following measurements of Cosines, Sines, and Tangent are based off angle theta. For example a Cosine ratio is adjacent side over hypotenuse side, so the ratio would be 17/20 from angle theta. Sines ratio would be 13/20 as side 13 is opposite form angle theta. Tangent Ration would be 13/17 as it is opposite over adjacent.
Law of Cosines: The law of cosines takes all three sides to find an angle, and two sides and one angle to find a missing side.
Solving for Angle
The first step to solve this problem would be to plug in the gives into the Cosine Angle formula which is; CosC=a2=b2-c2/2(a)(b)
The equation would be 9^2=17^2-20^2/2(9)(17)
The second step would be to use PEMDAS to factor everything into place, starting with the parenthesis following the exponents, multiplication,/division and finally addition and subtraction.
the equation will end up becoming -30/306. From this, you would have to plug this fraction into the cosine which will give you a decimal. Cos(-30/306)=(-0.098)
You will then have to use the inverse cosine function to end with the missing angle; Cos-1(-0.098)=95.62
Solving for Sides
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The first step to solving for your side is to plug in the given into the cosine formula, which goes as follows; c2=a2+b2-2(a)(b)cos(θ ). Therefore the formula for since c of the triangle would be c2=36^2+20^2-2(36)(20)cos(126). Step two would be to solve for PEMDAS. First, the parenthesis, then the exponents, followed by the subtraction. once everything is factored in you will need to square the sum. The sum of the square root will leave you with the totoal length of the side your solving for.
Law of Sines: The law of sines use one angle across from one of two side given to perform it equations.
Solving for angle
To solve for an angle, you will need to plug in the givens into the Sine Angle formula, which is; Sin(A)/a=Sin(B)/b=Sin(C)/c. Side a, b and angle A are given. The problem would best up as Sin(70)/12=Sin(θ )/6. To isolate Sin(θ ) you will need to multiply 6 to cancel it out. From step two you'll need to multiply 6 by Sin(70) to get your top denominator. you divide the sum of step three by 12 to get a decimal. You input this decimal into the inverse Sines function to come u with Sin(θ).
Solving for Side
Ex.
Step one would be to plug in the givens for the soling for side Sines equations which go as follows; a/Sin(A)=b/Sin(B)=c/sin(C). Step two would be to isolate the side form the whole equation. You'd have to multiply the Sin(θ) form the missing side giving you; Sin(B)(a)/Sin(A). You'd multiply the Sin(B) by (a) then divide it by Sin(A). This would give you the length of side b.
