Finding the Value of Trigonometric Expressions

We are given that \(\sin \theta = \frac{12}{13}\) and asked to find the value of the expression \(2\cot \theta + 13\cos \theta\). To solve this, we first need to find the values of \(\cos \theta\) and \(\cot \theta\) using the given information about \(\sin \theta\).

Using a Right-Angled Triangle

We can visualize this problem using a right-angled triangle. Recall that for a right-angled triangle, \(\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}\). Given \(\sin \theta = \frac{12}{13}\), we can consider a right-angled triangle where:

• The length of the side opposite to angle \(\theta\) is proportional to 12.
• The length of the hypotenuse is proportional to 13.

Let the opposite side be \(12k\) and the hypotenuse be \(13k\) for some positive constant \(k\). We need to find the length of the adjacent side. Using the Pythagorean theorem:

\((\text{Adjacent side})^2 + (\text{Opposite side})^2 = (\text{Hypotenuse})^2\)

\((\text{Adjacent side})^2 + (12k)^2 = (13k)^2\)

\((\text{Adjacent side})^2 + 144k^2 = 169k^2\)

\((\text{Adjacent side})^2 = 169k^2 - 144k^2\)

\((\text{Adjacent side})^2 = 25k^2\)

\(\text{Adjacent side} = \sqrt{25k^2} = 5k\) (Assuming \(k > 0\) and side length is positive).

Calculating cos θ and cot θ

Now that we have all three sides of the right-angled triangle, we can find \(\cos \theta\) and \(\cot \theta\).

Recall the definitions:

• \(\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\)
• \(\cot \theta = \frac{\text{Adjacent side}}{\text{Opposite side}}\)

Substituting the values:

\(\cos \theta = \frac{5k}{13k} = \frac{5}{13}\)

\(\cot \theta = \frac{5k}{12k} = \frac{5}{12}\)

(Note: This assumes \(\theta\) is in a quadrant where sine, cosine, and cotangent are positive, typically the first quadrant).

Evaluating the Expression 2cot θ + 13cos θ

Now we substitute the calculated values of \(\cot \theta\) and \(\cos \theta\) into the given expression:

Expression = \(2\cot \theta + 13\cos \theta\)

Substitute \(\cot \theta = \frac{5}{12}\) and \(\cos \theta = \frac{5}{13}\):

Expression = \(2 \times \left(\frac{5}{12}\right) + 13 \times \left(\frac{5}{13}\right)\)

Simplify the terms:

First term: \(2 \times \frac{5}{12} = \frac{10}{12} = \frac{5}{6}\)

Second term: \(13 \times \frac{5}{13} = 5\)

Add the simplified terms:

Expression = \(\frac{5}{6} + 5\)

To add these, find a common denominator, which is 6:

\(5 = \frac{5 \times 6}{1 \times 6} = \frac{30}{6}\)

Expression = \(\frac{5}{6} + \frac{30}{6} = \frac{5 + 30}{6} = \frac{35}{6}\)

Thus, the value of \(2\cot \theta + 13\cos \theta\) is \(\frac{35}{6}\).

Step-by-Step Calculation Summary

Revision Table: Trigonometric Ratios in a Right Triangle

Additional Information on Trigonometric Identities

Besides using a right-angled triangle, trigonometric identities can also be used to find missing ratios.

• The fundamental identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Given \(\sin \theta\), we can find \(\cos \theta\). \(\cos^2 \theta = 1 - \sin^2 \theta\) \(\cos \theta = \pm \sqrt{1 - \sin^2 \theta}\) In our case, \(\sin \theta = 12/13\). \(\cos \theta = \pm \sqrt{1 - \left(\frac{12}{13}\right)^2} = \pm \sqrt{1 - \frac{144}{169}} = \pm \sqrt{\frac{169 - 144}{169}} = \pm \sqrt{\frac{25}{169}} = \pm \frac{5}{13}\). We chose \(+\frac{5}{13}\) based on the assumption of \(\theta\) being in the first quadrant where cosine is positive.
• Once \(\sin \theta\) and \(\cos \theta\) are known, \(\cot \theta\) can be found using the identity \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). \(\cot \theta = \frac{5/13}{12/13} = \frac{5}{13} \times \frac{13}{12} = \frac{5}{12}\).

Both the triangle method and the identity method give the same values for \(\cos \theta\) and \(\cot \theta\), leading to the same final result for the expression.