Evaluate Trigonometric Expression at 30 Degrees

The question asks us to find the value of the expression \(5 \sin \theta - 2 \csc \theta\) when \(\theta = 30^\circ\). To solve this, we need to know the standard trigonometric values for an angle of \(30^\circ\) and understand the relationship between the sine and cosecant functions.

Understanding Sine and Cosecant

The sine function, denoted as \(\sin \theta\), is one of the fundamental trigonometric functions. The cosecant function, denoted as \(\csc \theta\), is the reciprocal of the sine function. This means:

\[\csc \theta = \frac{1}{\sin \theta}\]

Provided that \(\sin \theta \neq 0\).

Standard Trigonometric Values for 30 Degrees

For standard angles like \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\), the trigonometric values are commonly known or can be derived from special right triangles. For \(\theta = 30^\circ\):

• The value of \(\sin 30^\circ\) is \(1/2\).
• Since \(\csc \theta\) is the reciprocal of \(\sin \theta\), the value of \(\csc 30^\circ\) is the reciprocal of \(\sin 30^\circ\).

So, \(\csc 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{1/2} = 2\).

Evaluating the Expression

Now we substitute the values of \(\sin 30^\circ\) and \(\csc 30^\circ\) into the given expression \(5 \sin \theta - 2 \csc \theta\):

Substitute \(\theta = 30^\circ\):

\[5 \sin 30^\circ - 2 \csc 30^\circ\]

Substitute the values \(\sin 30^\circ = 1/2\) and \(\csc 30^\circ = 2\):

\[5 \left(\frac{1}{2}\right) - 2 (2)\]

Perform the multiplication:

\[\frac{5}{2} - 4\]

To subtract 4 from \(5/2\), we convert 4 into a fraction with a denominator of 2. Since \(4 = 8/2\):

\[\frac{5}{2} - \frac{8}{2}\]

Now subtract the numerators, keeping the common denominator:

\[\frac{5 - 8}{2} = \frac{-3}{2}\]

So, the value of the expression \(5 \sin \theta - 2 \csc \theta\) when \(\theta = 30^\circ\) is \(-3/2\).

Final Answer Calculation Steps

1. Identify the given expression: \(5 \sin \theta - 2 \csc \theta\).
2. Identify the value of \(\theta\): \(\theta = 30^\circ\).
3. Find the value of \(\sin 30^\circ\): \(\sin 30^\circ = 1/2\).
4. Find the value of \(\csc 30^\circ\) using \(\csc \theta = 1/\sin \theta\): \(\csc 30^\circ = 1/(1/2) = 2\).
5. Substitute these values into the expression: \(5(1/2) - 2(2)\).
6. Calculate the result: \(5/2 - 4 = 5/2 - 8/2 = -3/2\).

Revision Table: Trigonometry Basics

Additional Information: Reciprocal Trigonometric Functions

Besides sine and cosecant, there are other reciprocal trigonometric function pairs:

• Cosine (\(\cos \theta\)) and Secant (\(\sec \theta\)): \(\sec \theta = 1 / \cos \theta\).
• Tangent (\(\tan \theta\)) and Cotangent (\(\cot \theta\)): \(\cot \theta = 1 / \tan \theta\).

Understanding these reciprocal relationships is crucial for solving many trigonometric problems and simplifying expressions.

Remembering the standard trigonometric values for common angles (\(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\)) is very helpful for quickly evaluating trigonometric expressions like the one in this problem.