Which of the following identities is false for acute angles?
Answers
1) In ΔABC, ∠ABC = 2∠ACB
Let ∠ACB = x
⇒∠ABC = 2∠ACB = 2x
Given BP is bisector of ∠ABC
Hence ∠ABP = ∠PBC = x
Using the angle bisector theorem, that is,
the bisector of an angle divides the side opposite to it in the ratio of other two sides.
Hence, CB : BA= CP:PA.
2) Consider ΔABC and ΔAPB
∠ABC = ∠APB [Exterior angle property]
∠BCP = ∠ABP [Given]
∴ ΔABC ≈ ΔAPB [AA criterion]
∴fraction numerator space AB over denominator BP end fraction space equals space CA over CB[Corresponding sides of similar triangles are proportional.]
⇒ AB x BC = BP x CA
Therefore Identity 4 is FALSE for the acute angles.
Given:
Several identities which are listed below:
1.) ( Sin A + Sec A )² + ( Cos A + Cosec A )² = ( 1 + Sec A .Cosec A )²
2.) ( 1 - Sin A ) / ( 1 + Sin A ) = ( Sec A - Tan A )²
3.) ( 1 + Sin A ) / ( 1 - Sin A ) = ( Sec A + Tan A )²
4.) If A = 60° and B = 30° then Cos ( A - B ) = Cos A Cos B - Sin A Sin B
To Find:
Which of the given identities are FALSE for acute angle ( 0° < A < 90° )
Solution:
We can simply solve this numerical problem by using the following process.
Let us assume A = 30° for the first 3 identities and substitute in A so that whichever does not stand equal is the FALSE option.
Identity 1:
⇒ ( Sin A + Sec A )² + ( Cos A + Cosec A )² = ( 1 + Sec A .Cosec A )²
Substituting A = 30° in the above equation
⇒ ( Sin 30° + Sec 30° )² + ( Cos 30° + Cosec 30° )² = ( 1 + Sec 30° .Cosec 30° )²
Sin 30° = 0.5
Sec 30°= 1.155
Cos 30° = 0.866
Cosec 30° = 2
⇒ ( 0.5 + 1.155 )² + ( 0.866 + 2 )² = ( 1 + ( 1.155*2))²
⇒ ( 1.655 )² + ( 2.866 )² = ( 1 + 2.31 )²
⇒ 2.74 + 8.214 = 10.95
⇒ 10.95 = 10.95
LHS = RHS
∴ Identity 1 stands good for acute angles.
Identity 2:
⇒ ( 1 - Sin A ) / ( 1 + Sin A ) = ( Sec A - Tan A )²
Substitute A = 30°
⇒ ( 1 - Sin 30° ) / ( 1 + Sin 30° ) = ( Sec 30° - Tan 30° )²
Sin 30° = 0.5
Sec 30°= 1.155
Tan 30° = 0.577
⇒ ( 1 - 0.5 ) / ( 1 + 0.5 ) = ( 1.155 - 0.577 )²
⇒ 0.5 / 1.5 = 0.578²
⇒ 0.33 = 0.33
LHS = RHS
Hence Identity 2 stands good for acute angle.
Identity 3:
⇒ ( 1 + Sin A ) / ( 1 - Sin A ) = ( Sec A + Tan A )²
Substitute A = 30°
⇒ ( 1 + Sin 30° ) / ( 1 - Sin 30° ) = ( Sec 30° + Tan 30° )²
Sin 30° = 0.5
Sec 30°= 1.155
Tan 30° = 0.577
⇒ ( 1 + 0.5 ) / ( 1 - 0.5 ) = ( 1.155 + 0.577 )²
⇒ 1.5 / 0.5 = 1.732²
⇒ 3 = 3
LHS = RHS
Hence Identity 3 stands right for acute angles.
Identity 4:
⇒ Cos ( A - B ) = Cos A Cos B - Sin A Sin B
Substitute A = 60° and B = 30°
⇒ Cos ( 60° - 30° ) = Cos 60° Cos 30° - Sin 60° Sin 30°
⇒ Cos ( 30° ) = Cos 60° Cos 30° - Sin 60° Sin 30°
Cos 30° = 0.866
Sin 30° = 0.5
Cos 60° = 0.5
Sin 60° = 0.866
⇒ 0.866 = ( 0.5 * 0.866 ) - ( 0.5 * 0.866 )
⇒ 0.866 = 0
LHS ≠ RHS
Identity 4 does not stand right for the given angles.
Therefore Identity 4 is FALSE for the acute angles.
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