Question

In a ∆ABC, right-angled at C, if tan⁡ A= $\frac{1}{\surd 3}$, then find the value of sin⁡A sin⁡B + cos⁡A cos⁡B.

Answer 1

Step 1:

• First we need to determine the values of angles A and B and determine the value of their sine and cosine.
• We are given that tan A = $\frac{1}{\surd 3}$ and tan θ is $\frac{1}{\surd 3}$ for θ = 30º. Therefore,
  ∠A = 30º
• Also by angle sum property of a triangle,

∠A + ∠B + ​​​​​​​∠C ​​= 180º
Substituting ∠A = 30º and ∠C = 90º, We get ∠B = 60º

• Now, therefore,
• Sin A = sin 30º = $\frac{1}{2}$
• Sin B = sin 60º = $\frac{\sqrt{3}}{2}$
• Cos A = Cos 30º = $\frac{\sqrt{3}}{2}$
• Cos B = Cos 60º = $\frac{1}{2}$

Step 2:

• For the final step, we need to substitute the corresponding values in the given expression.

• The given expression is sinA sinB + cosA cosB.
  
  On substituting the respective values, we get

= $\frac{1}{2}\times \frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}\times \frac{1}{2}$
=$\frac{2\sqrt{3}}{4}$
= $\frac{\sqrt{3}}{2}$
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