Geometry/Right Triangles and Pythagorean Theorem

Right triangles Right triangles are triangles in which one of the interior angles is 90o. A 90o angle is called a right angle. Right triangles have special properties which make it easier to conceptualize and calculate their parameters in many cases. The side opposite of the right angle is called the hypotenuse. The sides adjacent to the right angle are the legs. When using the Pythagorean Theorem, the hypotenuse or its length is often labeled with a lower case c. The legs (or their lengths) are often labeled a and b. Image:Right_triangle_shows_hyp_legs.PNG Either of the legs can be considered a base and the other leg would be considered the height (or altitude), because the right angle automatically makes them perpendicular. If the lengths of both the legs are known, then by setting one of these sides as the base ( b ) and the other as the height ( h ), the area of the right triangle is very easy to calculate using this formula: Image:Right_triangle_bh.PNG(1/2) This is intuitively logical because another congruent right triangle can be placed against it so that the hypotenuses are the same line segment, forming a rectangle with sides having length b and width h. The area of the rectangle is b × h, so either one of the congruent right triangles forming it has an area equal to half of that rectangle. Image:Two right triangles in rectangle.PNG Right triangles can be neither equilateral, acute, nor obtuse triangles. Isosceles right triangles have two 45° angles as well as the 90° angle. All isosceles right triangles are similar since corresponding angles in isosceles right triangles are equal. If another triangle can be divided into two right triangles (see ../Triangle/), then the area of the triangle may be able to be determined from the sum of the two constituent right triangles. Also the Pythagorean theorem can be used for non right triangles. a2+b2=c2-2c Pythagorean Theorem For history regarding the Pythagorean Theorem, see Pythagorean theorem. The Pythagorean Theorem states that: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Let's take a right triangle as shown here and set c equal to the length of the hypotenuse and set a and b each equal to the lengths of the other two sides. Then the Pythagorean Theorem can be stated as this equation: Image:Pythagorean_triangle.PNG Using the Pythagorean Theorem, if the lengths of any two of the sides of a right triangle are known and it is known which side is the hypotenuse, then the length of the third side can be determined from the formula. Sine, Cosine, and Tangent for Right Triangles Sine, Cosine, and Tangent are all functions of an angle, which are useful in right triangle calculations. For an angle designated as θ, the sine function is abbreviated as sin θ, the cosine function is abbreviated as cos θ, and the tangent function is abbreviated as tan θ. For any angle θ, sin θ, cos θ, and tan θ are each single determined values and if θ is a known value, sin θ, cos θ, and tan θ can be looked up in a table or found with a calculator. There is a table listing these function values at the end of this section. For an angle between listed values, the sine, cosine, or tangent of that angle can be estimated from the values in the table. Conversely, if a number is known to be the sine, cosine, or tangent of an angle, then such tables could be used in reverse to find (or estimate) the value of a corresponding angle. These three functions are related to right triangles in the following ways: In a right triangle, the sine of a non-right angle equals the length of the leg opposite that angle divided by the length of the hypotenuse. the cosine of a non-right angle equals the length of the leg adjacent to it divided by the length of the hypotenuse. the tangent of a non-right angle equals the length of the leg opposite that angle divided by the length of the leg adjacent to it. For any value of θ where cos θ ≠ 0, . If one considers the diagram representing a right triangle with the two non-right angles θ1and θ2, and the side lengths a,b,c as shown here: Image:Right_triangle.PNG For the functions of angle θ1: Analogously, for the functions of angle θ2: Table of sine, cosine, and tangent for angles θ from 0 to 90° θ in degrees θ in radians sin θ cos θ tan θ 0 0 0.0 1.0 0.0 1 0.017453293 0.01745240 0.9998477 0.017455065 2 0.034906585 0.034899497 0.99939083 0.034920769 3 0.052359878 0.052335956 0.99862953 0.052407779 4 0.06981317 0.069756474 0.99756405 0.069926812 5 0.087266463 0.087155743 0.9961947 0.087488664 6 0.10471976 0.10452846 0.9945219 0.10510424 7 0.12217305 0.12186934 0.99254615 0.12278456 8 0.13962634 0.1391731 0.99026807 0.14054083 9 0.15707963 0.15643447 0.98768834 0.15838444 10 0.17453293 0.17364818 0.98480775 0.17632698 11 0.19198622 0.190809 0.98162718 0.19438031 12 0.20943951 0.20791169 0.9781476 0.21255656 13 0.2268928 0.22495105 0.97437006 0.23086819 14 0.2443461 0.2419219 0.97029573 0.249328 15 0.26179939 0.25881905 0.96592583 0.26794919 16 0.27925268 0.27563736 0.9612617 0.28674539 17 0.29670597 0.2923717 0.95630476 0.30573068 18 0.31415927 0.30901699 0.95105652 0.3249197 19 0.33161256 0.32556815 0.94551858 0.34432761 20 0.34906585 0.34202014 0.93969262 0.36397023 21 0.36651914 0.35836795 0.93358043 0.38386404 22 0.38397244 0.37460659 0.92718385 0.40402623 23 0.40142573 0.39073113 0.92050485 0.42447482 24 0.41887902 0.40673664 0.91354546 0.44522869 25 0.43633231 0.42261826 0.90630779 0.46630766 26 0.45378561 0.43837115 0.89879405 0.48773259 27 0.4712389 0.4539905 0.89100652 0.50952545 28 0.48869219 0.46947156 0.88294759 0.53170943 29 0.50614548 0.48480962 0.87461971 0.55430905 30 0.52359878 0.5 0.8660254 0.57735027 31 0.54105207 0.51503807 0.8571673 0.60086062 32 0.55850536 0.52991926 0.8480481 0.62486935 33 0.57595865 0.54463904 0.83867057 0.64940759 34 0.59341195 0.5591929 0.82903757 0.67450852 35 0.61086524 0.57357644 0.81915204 0.70020754 36 0.62831853 0.58778525 0.80901699 0.72654253 37 0.64577182 0.60181502 0.79863551 0.75355405 38 0.66322512 0.61566148 0.78801075 0.78128563 39 0.68067841 0.62932039 0.77714596 0.80978403 40 0.6981317 0.64278761 0.76604444 0.83909963 41 0.71558499 0.65605903 0.75470958 0.86928674 42 0.73303829 0.66913061 0.74314483 0.90040404 43 0.75049158 0.68199836 0.7313537 0.93251509 44 0.76794487 0.69465837 0.7193398 0.96568877 45 0.78539816 0.70710678 0.70710678 1.0 46 0.80285146 0.7193398 0.69465837 1.03553031 47 0.82030475 0.7313537 0.68199836 1.07236871 48 0.83775804 0.74314483 0.66913061 1.11061251 49 0.85521133 0.75470958 0.65605903 1.15036841 50 0.87266463 0.76604444 0.64278761 1.19175359 51 0.89011792 0.77714596 0.62932039 1.23489716 52 0.90757121 0.78801075 0.61566148 1.27994163 53 0.9250245 0.79863551 0.60181502 1.32704482 54 0.9424778 0.80901699 0.58778525 1.37638192 55 0.95993109 0.81915204 0.57357644 1.42814801 56 0.97738438 0.82903757 0.5591929 1.48256097 57 0.99483767 0.82367057 0.54463904 1.53986496 58 1.01229097 0.8480481 0.52991926 1.60033453 59 1.02974426 0.8571673 0.51503807 1.66427948 60 1.04719755 0.8660254 0.5 1.73205081 61 1.06465084 0.87461971 0.48480962 1.80404776 62 1.08210414 0.88294759 0.46947156 1.88072647 63 1.09955743 0.89100652 0.4539905 1.96261051 64 1.11701072 0.89879405 0.43837115 2.05030384 65 1.13446401 0.90630779 0.42261826 2.14450692 66 1.15191731 0.91354546 0.40673664 2.24603677 67 1.1693706 0.92050485 0.39073113 2.35585237 68 1.18682389 0.92718385 0.37460659 2.47508685 69 1.20427718 0.93358043 0.35836795 2.60508906 70 1.22173048 0.93969262 0.34202014 2.74747742 71 1.23918377 0.94551858 0.32556815 2.90421088 72 1.25663706 0.95105652 0.30901699 3.07768354 73 1.27409035 0.95630476 0.2923717 3.27085262 74 1.29154365 0.9612617 0.27563736 3.48741444 75 1.30899694 0.96592583 0.25881905 3.73205081 76 1.32645023 0.97029573 0.2419219 4.01078093 77 1.34390352 0.97437006 0.22495105 4.33147587 78 1.36135682 0.9781476 0.20791169 4.70463011 79 1.37881011 0.98162718 0.190809 5.14455402 80 1.3962634 0.98480775 0.17364818 5.67128182 81 1.41371669 0.98768834 0.15643447 6.31375151 82 1.43116999 0.99026807 0.1391731 7.11536972 83 1.44862328 0.99254615 0.12186934 8.14434643 84 1.46607657 0.9945219 0.10452846 9.51436445 85 1.48352986 0.9961947 0.087155743 11.4300523 86 1.50098316 0.99756405 0.069756474 14.3006663 87 1.51843645 0.99862953 0.052335956 19.0811367 88 1.53588974 0.99939083 0.034899497 28.6362533 89 1.55334303 0.9998477 0.01745240 57.2899616 90 1.57079633 1.0 0.0 not defined General rules for important angles: