Right Triangle Calculator With Square Roots

Hypotenuse Calculator.LetвЂ™s first look at some cases where we donвЂ™t know all the sides. In the triangle above, the hypotenuse is the side AB which is opposite the right angle, C. The other two sides of the triangle, AC and CB are referred to as the legs. (Only right triangles have a hypotenuse ). Includes converse tester.The hypotenuse is the largest side in a right triangle and is always opposite the right angle. We can use the Pythagorean theorem and properties of sines, cosines, and tangents to solve the triangle, that is, to find unknown parts in terms of known parts.Instantly calculates the missing side length of a right triangle from two known lengths, shows its work, and draws the triangle.

Now suppose we know the hypotenuse and one side, but have to find the other. Pythagorean Theorem This theorem shows that if you have a right triangle, the length of the hypotenuse is the square root of. We use the Pythagorean Theorem to solve for various values relating to the right-angled triangle.

Since sin AВ = a/c, we know cВ = a/sin AВ =В 15/sinВ 41. For example, if the side aВ =В 15 and the angle AВ =В 41В°, we can use a sine and a tangent to find the hypotenuse and the other side. UNIT 1: Square Roots and the Pythagorean Theorem.We might only know one side but we also know an angle. Calculate the area of each quadrilateral you made.

Then tan AВ = a/bВ =В 12.3/50.1В =В 0.2455. Suppose aВ =В 12.3 and bВ =В 50.1. Again, you use the trig functions, but in reverse. Whether you use a sine, cosine, or tangent depends on which side and angle you know.Inverse trig functions: arcsine, arccosine, and arctangentNow letвЂ™s look at the problem of finding angles if you know the sides. Also, tan AВ = a/b, so bВ = a/tan AВ = 15/tanВ 41В = 15/0.8693В = 17.256.

TheyвЂ™re pretty much redundant, but itвЂ™s worthwhile to know what they are in case you come across them. There are six ways of making ratios of two sides of a right triangle, and that gives the six functions:You can see by the listing that cotangent (abbreviated cot, or sometimes ctn) is the reciprocal of tangent, secant (abbreviated sec) is the reciprocal of cosine, and cosecant (abbreviated csc, or sometimes cosec) is the reciprocal of sine. You might ask, вЂњwhy six trig functions?вЂќ ItвЂ™s a kind of symmetry. In calculus, secant is frequently used.

There you have itвЂ“the secant is the line from the center of the circle to the tangent line AC. The angle AOB appears in the triangle COA as angle AOC, so sec AOBВ =В sec AOCВ =В hyp/adjВ = OC/OAВ = OC. By symmetry the tangent of the angle FOB is the line FG, but FOB is the complementary angle of AOB, hence, the cotangent of AOB is FG.Next, to interpret secants geometrically. Recall that its tangent is the line AC.

Find a and b.In the next few problems, the triangles arenвЂ™t right triangles, but you can solve them using what you know about right triangles.61. In a right triangle cВ =В 6 feet 3 inches and tan BВ =В 1.2. Express the results as common fractions.31. In each of the following right triangles of which two sides are given, compute the sin, cos, and tanВ of the angles A and B.

From the top of a lighthouse 175 feet high the angles of depression of the top and bottom of a flagpole are 23В° 17' and 42В° 38', respectively. What are the height and distance of the second building.207. From the top of a building 50 feet high the angles of elevation and depression of the top and bottom of another building are 19В° 41' and 26В° 34', respectively. If the side of an equilateral triangle is a, find the altitude, and the radii of the circumscribed and inscribed circles.202.

The top of a lighthouse is 230 feet above the sea. Take A and B at the same level.233. Find the altitude of the balloon. As a balloon passes between two points A and B, 2 miles apart, the angles of elevation of the balloon at these points are 27В° 19' and 41В° 45', respectively. Find the height of the tree.215.

There is more than one way to solve this problem. Unfortunately, tan B is the ratio of the two sides you donвЂ™t know, namely, b/a. So you only need the third side, which you can compute with the Pythagorean theorem, and then take ratios of two of the sides.31. You only need the sin, cos, and tanВ of the angles A and B you donвЂ™t need the angles themselves. What must be the elevation of an observer in order that he may be able to see an object on the earth thirty miles away? Assume the earth to be a smooth sphere.In each of the figures named in the next few problems the object is to express its area (i) in terms of the radius R, that is, the radius of the circumscribed circle, (ii) in terms of the apothem r, that is, the radius of the inscribed circle, and (iii) in terms of the side a.251.

Find B by tangents and c by the Pythagorean theorem.61. Since you have a and b, you can use tangents to find A and the Pythagorean theorem to find c.42. From that you can find cos B, and then a, and you can find sin B, and then b.34. From tan B, you can determine the angle B (use arctan). The Pythagorean theorem then gives 6.25 2В = a 2В +В 1.44 a 2, from which you can determine a, and then find b.Method 2.

By dropping perpendiculars from the another of the vertices of the equilateral triangle and using trig on the resulting small triangles, you can find the radii of these two circles.202. The circumscribed circle is the one passing through the three vertices, and the inscribed circle is the one inside tangent to the three sides. Drop a perpendicular from one vertex, say vertex C, and you get two congruent right triangles ACF and BCF, and you can find the length of that perpendicular, and thatвЂ™s the altitude of the equilateral triangle. An equilateral triangle ABC has three 60В° vertex angles. You can use tangents to find the two parts of the side AB and add them together.67. Although the triangle ABC is not a right triangle, it does break into two right triangles.

The further point is then xВ +В 65 feet from the tree. There are two unknowns: the height y of the tree, and the distance x of the nearer point to the tree. You can use it to find heights of inaccessible things. See, trig can be useful if youвЂ™re a lonely lighthouse keeper and donвЂ™t know what to do!214. Then the angle of elevation to the top of the other building will tell you how much higher it is than yours.207.

Various inverses of it have been used for centuries to compute the radius of the earth. A very interesting problem. Set up equations, and solve them.233. This is similar to 214, but in this problem, the balloon lies between the two points.

Set this problem up similar to 233, but different variables are known.251вЂ“255. The Pythagorean theorem the third side of the triangle.234. One side of a right triangle is r, the radius of the earth, and the hypotenuse is rВ + h where h is the height of the lighthouse. All you need here is the Pythagorean theorem.

You get 2 n little triangles. Draw lines from the center of the figure to the vertices and to the midpoints of of the sides. Let n be the number of sides on the regular polygon.