Maths-
General
Easy

Question

The sides of a triangle are sin invisible function application alpha comma cos invisible function application alpha spaceand √ left parenthesis 1 plus sin invisible function application alpha cos invisible function application alpha right parenthesis for some 0 less than alpha less than pi divided by 2.Then the greatest angle of the triangle is

  1. 60 to the power of 0
  2. 150 to the power of 0
  3. 120 to the power of ring operator
  4. 90 to the power of ring operator

hintHint:

In this question first we have to check which side is greatest and then we have to use Cosine rule. cos open parentheses C close parentheses equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction

The correct answer is: 120 to the power of ring operator


    In this question we are given the sides of triangle as sin open parentheses alpha close parentheses comma cos open parentheses alpha close parentheses comma square root of 1 plus sin open parentheses alpha close parentheses cos open parentheses alpha close parentheses end root
    Let the sides be a equals sin open parentheses alpha close parentheses comma space b equals cos open parentheses alpha close parentheses comma space c space equals square root of 1 plus sin open parentheses alpha close parentheses cos open parentheses alpha close parentheses end root
    It can be easily seen that the side c is the greatest. We know that the angle opposite to greatest side is the greatest.
    By using Cosine Rule
    cos open parentheses C close parentheses equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction
    =>cos open parentheses C close parentheses equals fraction numerator sin squared open parentheses alpha close parentheses plus cos squared open parentheses alpha close parentheses minus left parenthesis 1 plus sin open parentheses alpha close parentheses cos open parentheses alpha close parentheses right parenthesis over denominator 2 sin open parentheses alpha close parentheses cos open parentheses alpha close parentheses end fraction
    By using the trigonometric identity
    sin squared open parentheses x close parentheses plus cos squared open parentheses x close parentheses equals 1
    =>cos open parentheses C close parentheses equals fraction numerator 1 minus 1 minus sin open parentheses alpha close parentheses cos open parentheses alpha close parentheses over denominator 2 sin open parentheses alpha close parentheses cos open parentheses alpha close parentheses end fraction
    => cos open parentheses C close parentheses equals fraction numerator negative sin open parentheses alpha close parentheses cos open parentheses alpha close parentheses over denominator 2 sin open parentheses alpha close parentheses cos open parentheses alpha close parentheses end fraction
    => cos open parentheses C close parentheses equals fraction numerator negative 1 over denominator 2 end fraction
    =>C equals 120 degree
    Hence, the greatest angle is 120 degree

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