Question

The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is

1. nguyetanh
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
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