Prove that sin20×sin40×sin60×sin80=316

By K Balaji|Updated : November 12th, 2022

Given that x3-3x2-9x-5 is (x+1)(x+1)(x-5).

Consider LHS

Sin20×sin40×sin60×sin80

Taking common we get

=sin60[sin20×sin40×sin80]

Substituting the values and above equation can be written as

=32[sin20×sin(60–20)×sin(60+20)]

=32sin3(20)4∵sinAsin(60−A)sin(60+A)=14sin3A

=32sin604

=3232×4

=32×38

=316

=RHS

∴LHS=RHS

Hence proved

Sine

The sine function in trigonometry is the ratio of the hypotenuse's length to the opposite side's length in a right-angled triangle. To determine a right triangle's unknown angle or sides, utilise the sine function.

The sine function for any right triangle with an angle, let's say ABC, will be:

Sin α= Opposite/ Hypotenuse

Law of Sines Uses

In most cases, the rule of sines is employed to determine the triangle's elusive side or angle. If specific triangle measurement combinations are specified, this law can be applied.

  1. ASA Criteria: Determine the unknown side given two angles and the included side.
  2. AAS Criteria: To identify the unknown side given two angles and an excluded side.

As triangle congruence is demonstrated using the AAS and ASA procedures, these two conditions will yield a special solution.

Law of Sines in Real Life

  • The law of sines is really applied in engineering to determine the tilt angle.
  • In astronomy, it is used to calculate the separation between planets and stars.
  • Additionally, utilising the law of sines, navigational measurements are achievable.

Summary:-

Prove that sin20×sin40×sin60×sin80=316

It is proved that sin20×sin40×sin60×sin80=316

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