Evaluating Trigonometric Values and Checking Geometric Progression

The problem asks us to evaluate three trigonometric expressions for p, q, and r, and then check if two given statements about these values are correct. Let's evaluate each value step by step.

Calculating the Value of p

We are given \({\rm{p}} = \tan \left( { - \frac{{11{\rm{\pi }}}}{6}} \right)\).

We use the property that \(\tan(-x) = -\tan(x)\).

So, \({\rm{p}} = -\tan \left( \frac{{11{\rm{\pi }}}}{6}} \right)\).

Next, we simplify the angle \(\frac{{11{\rm{\pi }}}}{6}}\). We can write it as \(2\pi - \frac{\pi}{6}\).

Thus, \({\rm{p}} = -\tan \left( 2\pi - \frac{\pi}{6} \right)\).

Using the property that \(\tan(2\pi - x) = -\tan(x)\) (since \(2\pi - x\) is in the 4th quadrant where tan is negative), we get:

\(\tan \left( 2\pi - \frac{\pi}{6} \right) = -\tan \left( \frac{\pi}{6} \right)\).

Substituting this back into the expression for p:

\({\rm{p}} = - \left( -\tan \left( \frac{\pi}{6} \right) \right) = \tan \left( \frac{\pi}{6} \right)\).

The value of \(\tan \left( \frac{\pi}{6} \right)\) is \(\frac{1}{\sqrt{3}}\).

Therefore, \({\rm{p}} = \frac{1}{\sqrt{3}}\).

Calculating the Value of q

We are given \({\rm{q}} = \tan \left( {\frac{{21{\rm{\pi }}}}{4}} \right)\).

We simplify the angle \(\frac{{21{\rm{\pi }}}}{4}}\). We can write it as \(5\pi + \frac{{\rm{\pi }}}{4}\).

Thus, \({\rm{q}} = \tan \left( 5\pi + \frac{{\rm{\pi }}}{4} \right)\).

Using the property that \(\tan(n\pi + x) = \tan(x)\) for any integer n (since the period of tan is \(\pi\)), we have:

\({\rm{q}} = \tan \left( \frac{{\rm{\pi }}}{4} \right)\).

The value of \(\tan \left( \frac{\pi}{4} \right)\) is 1.

Therefore, \({\rm{q}} = 1\).

Calculating the Value of r

We are given \({\rm{r}} = \cot \left( {\frac{{283{\rm{\pi }}}}{6}} \right)\).

We simplify the angle \(\frac{{283{\rm{\pi }}}}{6}}\). We can write it as \(47\pi + \frac{{\rm{\pi }}}{6}\) (since \(283 \div 6 = 47\) with a remainder of 1, so \(283\pi/6 = (47 \times 6 + 1)\pi/6 = 47\pi + \pi/6\)).

Thus, \({\rm{r}} = \cot \left( 47\pi + \frac{{\rm{\pi }}}{6} \right)\).

Using the property that \(\cot(n\pi + x) = \cot(x)\) for any integer n (since the period of cot is \(\pi\)), we have:

\({\rm{r}} = \cot \left( \frac{{\rm{\pi }}}{6} \right)\).

The value of \(\cot \left( \frac{\pi}{6} \right)\) is \(\sqrt{3}\).

Therefore, \({\rm{r}} = \sqrt{3}\).

Summary of Calculated Values

We have found the values:

• \({\rm{p}} = \frac{1}{\sqrt{3}}\)
• \({\rm{q}} = 1\)
• \({\rm{r}} = \sqrt{3}\)

Analyzing Statement 1: The value of p × r is 2

We need to calculate the product \({\rm{p}} \times {\rm{r}}\).

\({\rm{p}} \times {\rm{r}} = \left( \frac{1}{\sqrt{3}} \right) \times \left( \sqrt{3} \right)\)

\({\rm{p}} \times {\rm{r}} = \frac{\sqrt{3}}{\sqrt{3}} = 1\).

The statement says that \({\rm{p}} \times {\rm{r}} = 2\). Since we calculated \({\rm{p}} \times {\rm{r}} = 1\), Statement 1 is incorrect.

Analyzing Statement 2: p, q and r are in G.P.

Three numbers a, b, and c are in Geometric Progression (G.P.) if the ratio of consecutive terms is constant, i.e., \(\frac{b}{a} = \frac{c}{b}\). This condition can be rewritten as \(b^2 = ac\).

In this case, the numbers are p, q, and r. For them to be in G.P., the condition is \({\rm{q}}^2 = {\rm{p}} \times {\rm{r}}\).

Let's check this condition using the values we calculated:

• \({\rm{q}}^2 = (1)^2 = 1\)
• \({\rm{p}} \times {\rm{r}} = \left( \frac{1}{\sqrt{3}} \right) \times \left( \sqrt{3} \right) = 1\)

Since \({\rm{q}}^2 = 1\) and \({\rm{p}} \times {\rm{r}} = 1\), the condition \({\rm{q}}^2 = {\rm{p}} \times {\rm{r}}\) is satisfied.

Therefore, p, q, and r are in G.P. Statement 2 is correct.

Conclusion

Based on our analysis:

• Statement 1: The value of \({\rm{p}} \times {\rm{r}}\) is 2. This is incorrect.
• Statement 2: p, q and r are in G.P. This is correct.

We are asked to select the correct answer using the code given. Option 1 states only 1 is correct, Option 2 states only 2 is correct, Option 3 states both 1 and 2 are correct, and Option 4 states neither 1 nor 2 is correct.

Since only Statement 2 is correct, the correct answer corresponds to the option that states "2 only".

Revision Table: Trigonometric Functions and G.P.

Additional Information: Understanding Trigonometric Periods and Identities

Understanding the periodic nature of trigonometric functions is crucial for evaluating them at large angles. The tangent and cotangent functions have a period of \(\pi\). This means that for any integer n, \(\tan(x + n\pi) = \tan(x)\) and \(\cot(x + n\pi) = \cot(x)\). This property allows us to reduce any angle to an equivalent angle within a single period (e.g., from 0 to \(\pi\) or -\(\pi/2\) to \(\pi/2\)) to find the function's value.

For negative angles, identities like \(\tan(-x) = -\tan(x)\) are useful. These identities are derived from the symmetry of the unit circle or the graphs of the functions.

A Geometric Progression is a sequence where the ratio between any term and the previous term is constant. For three terms p, q, and r to be in G.P., the common ratio must be the same: \(q/p = r/q\). Cross-multiplying gives \(q^2 = pr\), which is the condition we checked in Statement 2.