Evaluating Trigonometric Expressions with Tan

The problem asks us to find the value of a given expression involving trigonometric functions, specifically the tangent function, for particular angle values A and B.

We are given:

• Angle A = 60°
• Angle B = 30°

We need to evaluate the expression: \(\frac{(\tan A-\tan B)}{(1+\tan A \tan B)}\).

Recognizing the Tangent Difference Formula

The expression we need to evaluate looks very similar to a standard trigonometric identity, the tangent difference formula. The formula for the tangent of the difference of two angles, \((A-B)\), is:

$$\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$

Comparing the given expression with this formula, we can see they are identical in form. Therefore, the value of the given expression is equal to \(\tan(A-B)\) for the given values of A and B.

Calculating the Value using the Identity

Using the identity, we can substitute the given values of A and B into the formula:

Value of the expression = \(\tan(A-B) = \tan(60^\circ - 30^\circ)\)

First, calculate the difference between the angles:

\(60^\circ - 30^\circ = 30^\circ\)

So, the value of the expression is \(\tan(30^\circ)\).

We need to know the standard value of \(\tan(30^\circ)\).

The standard trigonometric values for common angles are often remembered or found in tables:

• \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)
• \(\tan(45^\circ) = 1\)
• \(\tan(60^\circ) = \sqrt{3}\)

Therefore, the value of \(\tan(30^\circ)\) is \(\frac{1}{\sqrt{3}}\).

Step-by-Step Calculation (Alternative Method)

We can also calculate the value by first finding \(\tan A\) and \(\tan B\) separately and then substituting them into the expression.

• A = 60°, so \(\tan A = \tan(60^\circ) = \sqrt{3}\)
• B = 30°, so \(\tan B = \tan(30^\circ) = \frac{1}{\sqrt{3}}\)

Now, substitute these values into the expression \(\frac{(\tan A-\tan B)}{(1+\tan A \tan B)}\):

$$\frac{(\tan 60^\circ-\tan 30^\circ)}{(1+\tan 60^\circ \tan 30^\circ)} = \frac{\left(\sqrt{3}-\frac{1}{\sqrt{3}}\right)}{\left(1+\sqrt{3} \cdot \frac{1}{\sqrt{3}}\right)}$$

Simplify the numerator:

$$\sqrt{3}-\frac{1}{\sqrt{3}} = \frac{\sqrt{3} \cdot \sqrt{3}}{\sqrt{3}} - \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} - \frac{1}{\sqrt{3}} = \frac{3-1}{\sqrt{3}} = \frac{2}{\sqrt{3}}$$

Simplify the denominator:

$$1+\sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1+1 = 2$$

Now, divide the simplified numerator by the simplified denominator:

$$\frac{\left(\frac{2}{\sqrt{3}}\right)}{2} = \frac{2}{\sqrt{3}} \div 2 = \frac{2}{\sqrt{3}} \times \frac{1}{2} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}}$$

Both methods give the same result, which is \(\frac{1}{\sqrt{3}}\).

Comparing with Options

Let's compare our calculated value with the given options:

• Option 1: 3
• Option 2: \(\frac{1}{\sqrt{3}}\)
• Option 3: \(\sqrt{3}\)
• Option 4: 1

Our calculated value is \(\frac{1}{\sqrt{3}}\), which matches Option 2.

Revision Table: Key Trigonometric Identities

Additional Information: Understanding Tan Values

The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. For common angles like 30°, 45°, and 60°, these ratios can be derived from special right-angled triangles (like the 30-60-90 triangle and the 45-45-90 triangle).

For a 30-60-90 triangle with hypotenuse 2, the side opposite 30° is 1, and the side opposite 60° is \(\sqrt{3}\). The adjacent side for 30° is \(\sqrt{3}\), and for 60° is 1.

• \(\tan(30^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{1}{\sqrt{3}}\)
• \(\tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3}\)

For a 45-45-90 triangle with legs of length 1, the hypotenuse is \(\sqrt{2}\).

• \(\tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{1}{1} = 1\)

Knowing these fundamental values and identities is crucial for solving problems involving trigonometric expressions and equations.