Answers

1. The sine of the smallest of the angles in the triangle is;

   SinA = √{(√5 – 1)/2}.

   What is geometric sequence?

   A unique kind of sequence called a geometric sequence has a constant ratio between every two succeeding terms. This ratio is regarded as one of the geometric sequence’s common ratios.

   • In other words, each phrase in a geometric series is multiplied by the a constant to produce the following term.
   • Therefore, a geometric series has the formula a, ar, ar², where an is the initial term as well as r is the sequence’s common ratio.
   • Either one positive or negative integer can be used to describe the common ratio.

   Now, according to the question;

   Consider right angled triangle ΔABC ; right angled at C.

   The side opposite to each vertices A,B,C are a, b, c respectively.

   Thus, by Pythagorean theorem,

   a² + b² = c²   (equation 1)

   By geometric sequence;

   a² = bc (say);

   Also, a/c = √(b/c)

   substitute  in equation 1

   b² + bc – c² = 0

   Divide equation by c².

   b²/c² + b/c – 1 = 0  (equation 2)

   Consider vertex B.

   The sine of angle B; sinB = Perpendicular/Hypotenuse

   SinB = b/c = t (say)

   Substitute b/c by t in equation 2

   t² + t -1 = 0

   Calculate the roots of the equation by quadratic formula;

   t = (√5 – 1)/2 and (-√5 – 1)/2 (negative value is not possible for side)

   Thus, t = (√5 – 1)/2

   Also SinA = a/c = √(b/c)

   SinA = √{(√5 – 1)/2}

   Therefore, the sine of the smallest of the angles in the triangle is;

   SinA = √{(√5 – 1)/2}.

   To know more about geometric sequence, here

   https://brainly.com/question/1662572

   #SPJ4

   Reply