Finding Cosine Value from Tangent Equation
We are given the equation $a \tan \alpha = b$. Our goal is to find the value of $\cos \alpha$.
Step 1: Express tan α in terms of a and b
From the given equation, we can isolate $\tan \alpha$: $ \tan \alpha = \frac{b}{a} $
Step 2: Relate tan α to a right-angled triangle
Recall that in a right-angled triangle, $\tan \alpha = \frac{\text{opposite side}}{\text{adjacent side}}$. We can associate $b$ with the length of the opposite side and $a$ with the length of the adjacent side relative to angle $\alpha$.
Step 3: Calculate the hypotenuse
Using the Pythagorean theorem, where hypotenuse$^2$ = opposite$^2$ + adjacent$^2$: $ \text{hypotenuse}^2 = b^2 + a^2 $ $ \text{hypotenuse} = \sqrt{a^2 + b^2} $
Step 4: Calculate cos α
Recall that $\cos \alpha = \frac{\text{adjacent side}}{\text{hypotenuse}}$. Substituting the values we found: $ \cos \alpha = \frac{a}{\sqrt{a^2 + b^2}} $
Conclusion
Therefore, the value of $\cos \alpha$ is $\frac{a}{\sqrt{a^2 + b^2}}$. This matches Option 3.
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