Understanding the Trigonometry Problem

The question asks us to find the value of $\text{tan } \theta - \text{Cot } \theta$ given that $\text{Sin } \theta = \(\frac{4}{5}\)$. This is a common trigonometry problem that involves using the definitions of trigonometric ratios and the Pythagorean theorem.

Using the Given Sine Value

We are given that $\text{Sin } \theta = \(\frac{4}{5}\)$. Recall the definition of Sine in a right-angled triangle:

$\text{Sin } \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}$

So, if we consider a right-angled triangle with angle $\theta$, the ratio of the opposite side to the hypotenuse is 4:5. We can assume the opposite side is 4 units and the hypotenuse is 5 units for simplicity in calculation.

Finding the Adjacent Side using Pythagorean Theorem

To find the values of $\text{tan } \theta$ and $\text{Cot } \theta$, we need to know the length of the adjacent side. We can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (opposite and adjacent sides).

$\text{Adjacent}^2 + \text{Opposite}^2 = \text{Hypotenuse}^2$

Substituting the known values:

$\text{Adjacent}^2 + (4)^2 = (5)^2$

$\text{Adjacent}^2 + 16 = 25$}

Now, subtract 16 from both sides to find $\text{Adjacent}^2$:

$\text{Adjacent}^2 = 25 - 16$

$\text{Adjacent}^2 = 9$}

Taking the square root of both sides to find the length of the adjacent side:

$\text{Adjacent} = \sqrt{9}$

$\text{Adjacent} = 3$ units

So, the three sides of the right-angled triangle are: Opposite = 4, Adjacent = 3, Hypotenuse = 5.

Calculating Tan and Cot Values

Now we can find the values of $\text{tan } \theta$ and $\text{Cot } \theta$ using their definitions:

$\text{tan } \theta = \frac{\text{Opposite side}}{\text{Adjacent side}}$

$\text{tan } \theta = \frac{4}{3}$

$\text{Cot } \theta = \frac{\text{Adjacent side}}{\text{Opposite side}}$

$\text{Cot } \theta = \frac{3}{4}$}

Finding the Value of tan θ - Cot θ

Finally, we need to calculate $\text{tan } \theta - \text{Cot } \theta$:

$\text{tan } \theta - \text{Cot } \theta = \frac{4}{3} - \frac{3}{4}$}

To subtract these fractions, we need a common denominator. The least common multiple of 3 and 4 is 12.

Convert $\frac{4}{3}$ to a fraction with a denominator of 12:

$\frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12}$

Convert $\frac{3}{4}$ to a fraction with a denominator of 12:

$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

Now perform the subtraction:

$\text{tan } \theta - \text{Cot } \theta = \frac{16}{12} - \frac{9}{12} = \frac{16 - 9}{12}$}

$\text{tan } \theta - \text{Cot } \theta = \frac{7}{12}$}

The value of $\text{tan } \theta - \text{Cot } \theta$ is $\(\frac{7}{12}\)$.

Summary of Steps

1. Identify the given trigonometric ratio ($\text{Sin } \theta$).
2. Use the definition of $\text{Sin } \theta$ to identify the ratio of the opposite side to the hypotenuse in a right-angled triangle.
3. Use the Pythagorean theorem ($\text{Adjacent}^2 + \text{Opposite}^2 = \text{Hypotenuse}^2$) to find the length of the adjacent side.
4. Use the definitions of $\text{tan } \theta$ ($\frac{\text{Opposite}}{\text{Adjacent}}$) and $\text{Cot } \theta$ ($\frac{\text{Adjacent}}{\text{Opposite}}$) to find their values.
5. Subtract $\text{Cot } \theta$ from $\text{tan } \theta$ by finding a common denominator for the fractions.
6. Simplify the result to get the final answer.

Revision Table: Key Trigonometric Ratios

Understanding the basic trigonometric ratios is crucial for solving such problems. Here's a quick reference:

• $\text{Sin } \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• $\text{Cos } \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• $\text{Tan } \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\text{Sin } \theta}{\text{Cos } \theta}$
• $\text{Cot } \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{1}{\text{Tan } \theta}$
• $\text{Sec } \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{1}{\text{Cos } \theta}$
• $\text{Cosec } \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{1}{\text{Sin } \theta}$

Additional Information: Pythagorean Identities

Beyond the basic ratios, there are fundamental identities derived from the Pythagorean theorem that are very useful in trigonometry:

• $\text{Sin}^2 \theta + \text{Cos}^2 \theta = 1$
• $\text{Tan}^2 \theta + 1 = \text{Sec}^2 \theta$
• $\text{Cot}^2 \theta + 1 = \text{Cosec}^2 \theta$

These identities can often be used to simplify expressions or find unknown trigonometric values if one value is given, sometimes providing an alternative path to the solution compared to directly using the sides of a right-angled triangle.