4Solution of Right-angled Triangle Session ObjectiveSolution of Right-angled TriangleSolution of a Oblique Triangle(a) When three sides are given(b) When three angles are given(c) When two sides and the included anglebetween them are givenTwo angles and one of the corresponding sides are given(e) When two sides and an angle opposite to one of them is given
5IntroductionA triangle has three sides and three angles. If three parts of a triangle, at least one of which is side, are given then other three parts can be uniquely determined.Finding other unknown parts, when three parts are known is called ‘solution of triangle’.
6Solution of Right-angled Triangle (a) Given a side and an acute angleLet the angle A (acute) and side c of aright angle at C be givencbaCABor b = c sinB, a = c cosB
7Solution of Right-angled Triangle (b) Given two sidesLet a and b are the sides of C is the right angle. Then we can find the remaining angles and sides by the following waycabCBA
8Solution of a Oblique Triangle (a) When three sides are givenIf the given data is in sine, use the following formulaeIf the given data is in cosine, use the following formulae.
9Solution of a Oblique Triangle If the given data is in tangent,then we use(iv) If the lengths of the sides a, b and c are small, the angle of triangle can also be obtained by cosine rule.(v) For logarithmic computation, we define the following.
10Solution of a Oblique Triangle (b) When three angles are given:In this case, the sides cannot be determined uniquely. Only ratio of the sides can be determined by sine rule and hence there will be infinite number of such triangles.
11Solution of a Oblique Triangle (c) When two sides and the included angle between them are given:If two sides b and c and the included angle A are given, then (B – C) can be found by using the following formula:If b < c, then we use
12Solution of a Oblique Triangle Two angles and one of thecorresponding sides are given:The value of the other side and remainingangle can be found by
13Solution of a Oblique Triangle (e) When two sides and an angle opposite to one of them is given:In this case, either no triangle or one triangle or two triangles are possible depending on the given parts.Therefore, this case is known as ambiguous case.Let a, b and the angle A are given.
14Solution of a Oblique Triangle This is a quadratic equation in c.Let c1 and c2 be two values of c
15Solution of a Oblique Triangle Case I: When a < b sinAHence, from (i) and (ii), become imaginary.No triangle is possible.
16Solution of a Oblique Triangle Case II: When a = b sinA From (i) and (ii),But will be positive when A is acute angle. In this case, only one triangle is possible provided is acute.
17Solution of a Oblique Triangle Also a = b sinAThe triangle is right-angled in this case.Case III: When a > b sinA From (i) and (ii), are real.
18Solution of a Oblique Triangle But triangle is possible only when are positive. For this we have to consider the following cases:(a) When a > bHence only one triangleis possible.
19Solution of a Oblique Triangle (b) When a = bHence, only one triangle is possible.
20Solution of a Oblique Triangle (c) When a < b Two triangles are possible. (Ambiguous case)Thus, when a, b and are given, two triangles are possible when a > b sinA and a < b [For ambiguous case]