How to find the perimeter of a triangle using trigonometry

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How to find the perimeter of a triangle using trigonometry

Trigonometry can be used to find the area of a triangle when you know the length of two sides and the included angle. home / math / right triangle calculator Please provide 2 values below to calculate the other values of a right triangle. If radians are selected as the angle unit, it can take values such as pi/3, pi/4, etc. RelatedTriangle Calculator | Pythagorean Theorem Calculator Right triangle A right triangle is a type of triangle that has one angle that measures 90?. Right triangles, and the relationships between their sides and angles, are the basis of trigonometry. In a right triangle, the side that is opposite of the 90? angle is the longest side of the triangle, and is called the hypotenuse. The sides of a right triangle are commonly referred to with the variables a, b, and c, where c is the hypotenuse and a and b are the lengths of the shorter sides. Their angles are also typically referred to using the capitalized letter corresponding to the side length: angle A for side a, angle B for side b, and angle C (for a right triangle this will be 90?) for side c, as shown below. In this calculator, the Greek symbols (alpha) and (beta) are used for the unknown angle measures. h refers to the altitude of the triangle, which is the length from the vertex of the right angle of the triangle to the hypotenuse of the triangle. The altitude divides the original triangle into two smaller, similar triangles that are also similar to the original triangle. If all three sides of a right triangle have lengths that are integers, it is known as a Pythagorean triangle. In a triangle of this type, the lengths of the three sides are collectively known as a Pythagorean triple. Examples include: 3, 4, 5; 5, 12, 13; 8, 15, 17, etc. Area and perimeter of a right triangle are calculated in the same way as any other triangle. The perimeter is the sum of the three sides of the triangle and the area can be determined using the following equation: Special Right Triangles 30?-60?-90? triangle: The 30?-60?-90? refers to the angle measurements in degrees of this type of special right triangle. In this type of right triangle, the sides corresponding to the angles 30?-60?-90? follow a ratio of 1:3:2. Thus, in this type of triangle, if the length of one side and the side's corresponding angle is known, the length of the other sides can be determined using the above ratio. For example, given that the side corresponding to the 60? angle is 5, let a be the length of the side corresponding to the 30? angle, b be the length of the 60? side, and c be the length of the 90? side.: Angles: 30?: 60?: 90? Ratio of sides: 1:3:2 Side lengths: a:5:c Then using the known ratios of the sides of this special type of triangle: As can be seen from the above, knowing just one side of a 30?-60?-90? triangle enables you to determine the length of any of the other sides relatively easily. This type of triangle can be used to evaluate trigonometric functions for multiples of /6. 45?-45?-90? triangle: The 45?-45?-90? triangle, also referred to as an isosceles right triangle, since it has two sides of equal lengths, is a right triangle in which the sides corresponding to the angles, 45?-45?-90?, follow a ratio of 1:1:2. Like the 30?-60?-90? triangle, knowing one side length allows you to determine the lengths of the other sides of a 45?-45?-90? triangle. Angles: 45?: 45?: 90? Ratio of sides: 1:1:2 Side lengths: a:a:c Given c= 5: 45?-45?-90? triangles can be used to evaluate trigonometric functions for multiples of /4. A Trigonometric Formula for the Area of a Triangle: The general formula for the area of a triangle is well known. While the formula shows the letters b and h, it is actually the pattern of the formula that is important. The area of a triangle equals ? the length of one side times the height drawn to that side (or an extension of that side). General Formula for Area of Triangle: b = length of a side (base) h = height draw to that side The area of ABC can be expressed as: where a represents the side (base) and h represents the height drawn to that side. Using trigonometry, let's take another look at this diagram. In the right triangle CDA, we can state that: The height, h, of the triangle can be expressed as b sin C. Substituting this new expression for the height, h, into the general formula for the area of a triangle gives: where a and b can be any two sides and C is the included angle. The area of a triangle can be expressed using the lengths of two sides and the sine of the included angle. Area = ? ab sin C. You may see this referred to as the SAS formula for the area of a triangle. With this new formula, we no longer have to rely on finding the altitude (height) of a triangle in order to find its area. Now, if we know two sides and the included angle of a triangle, we can find the area of the triangle. This is a valuable new formula! Given the triangle at the right, find its area. Express the answer to the nearest hundredth of a square unit. When using your graphing calculator, be sure you are in DEGREE mode, or using the degree symbol. Let a = ST, b = RT, and C = RTS. Given the parallelogram shown at the right, find its area to the nearest square unit. The diagonal of a parallelogram divides it into two congruent triangles. So the total area of the parallelogram will be TWICE the area of one of the triangles formed by the diagonal. This example shows that by doubling the triangle area formula, we have created a formula for finding the area of a parallelogram, given 2 adjacent sides (a and b) and the included angle, C. Area of Parallelogram Let a = PS, b - RS, and C =PSR. Given the parallelogram shown at the right, find its EXACT area. If a question asks for an EXACT answer, do not use your calculator to find the sin 60? since it will be a rounded value. To get an EXACT value for sin 60?, use the 30?-60?-90? special triangle which gives the sin 60? to be . Notice that we are using the formula for the area of a parallelogram we discovered in Example 2. Let a = AD, b = AB, and C = BAD. For help with this formula on your calculator, click here. Deriving this formula: NOTE: The Common Core Standard G.SRT.9 states "Derive the formula A = ?ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side." This statement can be interpreted as applying only to acute triangles. This site will, however, examine both "acute" and "obtuse" triangles in deriving the formula. Case 1: Acute Triangle If this formula truly works (and it does!), we should be able to apply the formula using any angle in the triangle. So, when attempting to "derive" this formula, we should show that it can be "developed" using any (and every) angle in the triangle. In the example shown above, we developed the formula using acute C. The same approach can be used to establish the relationship using acute B: . When A is a third acute angle, we can draw another internal altitude (height) and apply this same approach a third time, getting: . Case 2: Obtuse Triangle Can we still develop this formula if A is an obtuse angle? The answer is "yes", but it will require more work and some more trigonometric information. We will take a brief look at what is involved when A is an obtuse angle, but these concepts will be more fully developed in upcoming courses. Note: to maintain the use of a single letter to represent the angle in our formula, we will be referring to BAC in the diagram below, as A. When A is an obtuse angle, the altitude drawn from C or B will be outside of the triangle. Draw the altitude from C to the line containing the opposite side. CAE is a right triangle, but unfortunately it does not contain A that we need for our formula. We know, however, that CAE is supplementary to A, since they form a linear pair. We can state that mCAE = 180 - mA and from CAE that . If we apply a trigonometric fact that sinA = sin(180 - mA), we can substitute and get: (After multiplying both sides of the first equation by b.) Now, substitution into the general formula for the area of a triangle will give us our desired formula: . sinA = sin (180 - mA) Remember that the functions of sine, cosine, and tangent are defined only for acute angles in a right triangle. So, how do we find the sine of an obtuse angle? We cannot use the sides of the triangle to find sinBAC because the angle does not reside in a right triangle. We can, however, find sinBAD which deals with an acute angle in a right triangle. BAD is the supplement of BAC since they form a linear pair. The sine of an obtuse angle is defined to be the sine of the supplement of the angle. Thus, sinA = sin (180 - mA). On your graphing calculator, sin(50?) = 0.7660444431 and sin(130?) = 0.7660444431show this fact to be true. These angles are supplementary since 50? + 130? = 180?. WHY does sinA = sin (180 - mA)? This topic will be explored in more detail in upcoming courses. To understand "why" this relationship is true, we need a coordinate grid. Right triangle DEF is drawn in quadrant I, as shown. If we draw an angle of 130?, and drop a perpendicular to the x-axis from point H where DH = DF, we will create a reflection of DEF over the y-axis. This reflected triangle (DGH) is congruent to DEF and both triangles have the same lengths for their sides opposite the 50?. It should be noted that both opposite sides deal with positive y-values (designating direction above the x-axis). 50? and 130? are supplementary. When dealing with obtuse angles (such as 130?), the corresponding acute angle (50?) is used to determine the sine, cosine or tangent of that obtuse angle. This corresponding acute angle is called a "reference angle". NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use".

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