If cos $\theta$ = $\frac{\text{x}}{\text{y}}$, find tan $\theta$ in terms of x and y

Analysis

To solve this problem, we need to use the basic trigonometric ratios in a right-angled triangle. We are given the value of $\cos \theta$ and asked to find $\tan \theta$.

Concept: SOH-CAH-TOA

In a right-angled triangle:

• $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

We are given $\cos \theta = \frac{x}{y}$. This means:

• Adjacent side = $x$
• Hypotenuse = $y$

To find $\tan \theta$, we need the Opposite side. We can find this using the Pythagorean theorem:

Step-by-Step Solution

1. Find the Opposite side: Rearrange the Pythagorean theorem to solve for the Opposite side: $$\text{Opposite}^2 = y^2 - x^2$$ $$\text{Opposite} = \sqrt{y^2 - x^2}$$

2. Calculate $\tan \theta$: Using the formula $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$: $$\tan \theta = \frac{\sqrt{y^2 - x^2}}{x}$$

Conclusion

Comparing our result to the given options, the correct expression is $\frac{\sqrt{y^2 - x^2}}{x}$. This matches option A.

Exam Tip

Whenever you are given one trigonometric ratio (like $\sin, \cos,$ or $\tan$) and asked for another, always sketch a quick right-angled triangle. Label the sides based on the ratio given, use Pythagoras to find the missing third side, and then write down the required ratio. This visual method prevents silly mistakes with square roots!