Question

If alpha + beta = 90 degree show that the maximum value of cos alpha cos beta is 1/2

Answer 1

Answer:

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Step-by-step explanation:

Maximum and minimum value of sinα sinβ is 1/2 and -1/2 respectively.

Step-by-step explanation:

Given: \alpha+\beta=90^{\circ}=\frac{\pi}{2}α+β=90

∘

=

2

π

To find: Maximum and Minimum Value of sin\,\alpha\:\,sin\,\betasinαsinβ

Consider,

sin\,\alpha\:\,sin\,\betasinαsinβ

Using Complementary angle rule,

=sin\,\alpha\:\,cos\,\alpha=sinαcosα

=\frac{2}{2}\times sin\,\alpha\:\,cos\,\alpha=

2

2

×sinαcosα

=\frac{2sin\,\alpha\:\,cos\,\alpha}{2}=

2

2sinαcosα

Using Half Angle formula of trigonometry,

=\frac{sin\,2\alpha}{2}=

2

sin2α

Range of the sin function is [ -1 , 1 ]

⇒ Maximum Value = 1

⇒ Minimum Value = 1

So, Maximum Value of sinα sinβ = \frac{1}{2}

2

1

Minimum Value of sinα sinβ = \frac{-1}{2}

2

−1

Therefore, Maximum and minimum value of sinα sinβ is 1/2 and -1/2 respectively.

Answer 2

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