Finding the Value of a Trigonometric Expression

The question asks us to find the value of the expression \( \frac{{\sin \,\theta \, + \,\cos \,\theta }}{{\sin \,\theta \, - \,\cos \,\theta }} \) when \( \theta = 45^\circ \). To solve this, we need to know the values of \( \sin 45^\circ \) and \( \cos 45^\circ \).

Trigonometric Values for \( \theta = 45^\circ \)

At \( \theta = 45^\circ \), the sine and cosine values are equal. Specifically:

• \( \sin 45^\circ = \frac{1}{{\sqrt 2 }} \)
• \( \cos 45^\circ = \frac{1}{{\sqrt 2 }} \)

These values are fundamental in trigonometry and are often learned together with values for \( 30^\circ \) and \( 60^\circ \). Let's summarize some common angle values in a table.

Evaluating the Expression at \( \theta = 45^\circ \)

Now we substitute the values of \( \sin 45^\circ \) and \( \cos 45^\circ \) into the given expression:

Expression = \( \frac{{\sin \,45^\circ \, + \,\cos \,45^\circ }}{{\sin \,45^\circ \, - \,\cos \,45^\circ }} \)

Substitute the values:

Expression = \( \frac{{\frac{1}{{\sqrt 2 }} + \frac{1}{{\sqrt 2 }}}}{{\frac{1}{{\sqrt 2 }} - \frac{1}{{\sqrt 2 }}}} \)

Simplifying the Numerator and Denominator

Let's simplify the numerator first:

Numerator = \( \frac{1}{{\sqrt 2 }} + \frac{1}{{\sqrt 2 }} = \frac{1+1}{{\sqrt 2 }} = \frac{2}{{\sqrt 2 }} \)

We can simplify \( \frac{2}{{\sqrt 2 }} \) by multiplying the numerator and denominator by \( {\sqrt 2 } \):

\( \frac{2}{{\sqrt 2 }} \times \frac{{\sqrt 2 }}{{\sqrt 2 }} = \frac{{2{\sqrt 2 }}}{2} = {\sqrt 2 } \)

Now, let's simplify the denominator:

Denominator = \( \frac{1}{{\sqrt 2 }} - \frac{1}{{\sqrt 2 }} = 0 \)

Final Calculation

Now, substitute the simplified numerator and denominator back into the expression:

Expression = \( \frac{{\text{Numerator}}}{{\text{Denominator}}} = \frac{{\sqrt 2 }}{0} \)

Division by zero is undefined. In this context, when the numerator is a non-zero number and the denominator is zero, the value of the expression tends towards infinity (\( \infty \)).

Therefore, the value of the expression \( \frac{{\sin \,\theta \, + \,\cos \,\theta }}{{\sin \,\theta \, - \,\cos \,\theta }} \) when \( \theta = 45^\circ \) is \( \infty \).

Revision Table: Key Concepts in Trigonometry

Additional Information on Trigonometric Functions

Trigonometric functions like sine and cosine are periodic functions defined on the unit circle. Their values for standard angles are derived from geometry (like the properties of 45-45-90 and 30-60-90 triangles).

The expression \( \frac{{\sin \,\theta \, + \,\cos \,\theta }}{{\sin \,\theta \, - \,\cos \,\theta }} \) can also be manipulated using trigonometric identities. Dividing the numerator and denominator by \( \cos \theta \) (assuming \( \cos \theta \ne 0 \)) gives:

\( \frac{{\frac{{\sin \,\theta }}{{\cos \,\theta }} + \frac{{\cos \,\theta }}{{\cos \,\theta }}}}{{\frac{{\sin \,\theta }}{{\cos \,\theta }} - \frac{{\cos \,\theta }}{{\cos \,\theta }}}} = \frac{{\tan \,\theta \, + \,1}}{{\tan \,\theta \, - \,1}} \)

For \( \theta = 45^\circ \), \( \tan 45^\circ = 1 \). Substituting this into the simplified expression gives:

\( \frac{{1\, + \,1}}{{1\, - \,1}} = \frac{2}{0} \)

This confirms our previous result that the expression is undefined or \( \infty \) at \( \theta = 45^\circ \).

It's important to remember that division by zero indicates that the expression is not defined at that specific value of \( \theta \).