Finding the Value of θ in a Trigonometric Equation

The question asks us to find the value of the angle θ given the trigonometric equation:

\[ \tan \theta = \frac {\cos 17^\circ - \sin 17^\circ}{\cos 17^\circ + \sin 17^\circ} \]

To solve this, we need to simplify the right-hand side of the equation. A common technique when dealing with expressions involving \(\cos A \pm \sin A\) is to divide both the numerator and the denominator by either \(\cos A\) or \(\sin A\).

Let's divide the numerator and the denominator of the right-hand side by \(\cos 17^\circ\):

\[ \frac {\frac{\cos 17^\circ}{\cos 17^\circ} - \frac{\sin 17^\circ}{\cos 17^\circ}}{\frac{\cos 17^\circ}{\cos 17^\circ} + \frac{\sin 17^\circ}{\cos 17^\circ}} \]

Now, we use the identity \(\frac{\sin x}{\cos x} = \tan x\). So, \(\frac{\sin 17^\circ}{\cos 17^\circ} = \tan 17^\circ\).

Substituting this into the expression, we get:

\[ \frac {1 - \tan 17^\circ}{1 + \tan 17^\circ} \]

This expression looks similar to the tangent subtraction formula, which is \(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\).

We know that \(\tan 45^\circ = 1\). We can substitute \(1\) with \(\tan 45^\circ\) in the numerator and in the denominator (as the coefficient of \(\tan 17^\circ\)):

\[ \frac {\tan 45^\circ - \tan 17^\circ}{1 + \tan 45^\circ \tan 17^\circ} \]

Now, this expression perfectly matches the tangent subtraction formula with \(A = 45^\circ\) and \(B = 17^\circ\). So, the right-hand side simplifies to:

\[ \tan(45^\circ - 17^\circ) \]

Calculate the difference in the angle:

\[ 45^\circ - 17^\circ = 28^\circ \]

So, the right-hand side of the original equation simplifies to \(\tan 28^\circ\). The original equation was \(\tan \theta = \frac {\cos 17^\circ - \sin 17^\circ}{\cos 17^\circ + \sin 17^\circ}\), which now becomes:

\[ \tan \theta = \tan 28^\circ \]

For the general solution, \(\theta = n\pi + 28^\circ\), where \(n\) is an integer. However, given the options in degrees and typically seeking the principal value in such problems, we can conclude that:

\[ \theta = 28^\circ \]

Step-by-Step Solution to Find θ

1. Start with the given equation: \(\tan \theta = \frac {\cos 17^\circ - \sin 17^\circ}{\cos 17^\circ + \sin 17^\circ}\).
2. Divide the numerator and denominator of the right side by \(\cos 17^\circ\).
3. Use the identity \(\frac{\sin x}{\cos x} = \tan x\) to simplify the expression to \(\frac{1 - \tan 17^\circ}{1 + \tan 17^\circ}\).
4. Recognize that \(1 = \tan 45^\circ\). Substitute \(1\) with \(\tan 45^\circ\) in the expression: \(\frac{\tan 45^\circ - \tan 17^\circ}{1 + \tan 45^\circ \tan 17^\circ}\).
5. Apply the tangent subtraction formula \(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\) with \(A = 45^\circ\) and \(B = 17^\circ\).
6. The right side simplifies to \(\tan(45^\circ - 17^\circ) = \tan 28^\circ\).
7. Equate \(\tan \theta = \tan 28^\circ\).
8. The principal value of \(\theta\) is \(28^\circ\).

Key Trigonometric Identity Used

The crucial identity used in solving this problem is the tangent subtraction formula:

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

We also used the identity \(\tan 45^\circ = 1\).

Revision Table: Trigonometric Formulas

Additional Information: Handling Similar Trigonometric Expressions

Expressions like \(\frac{\cos A - \sin A}{\cos A + \sin A}\) or \(\frac{\cos A + \sin A}{\cos A - \sin A}\) frequently appear in trigonometry problems. The standard approach to simplify them is to divide the numerator and denominator by either \(\cos A\) or \(\sin A\).

• Dividing by \(\cos A\) leads to expressions involving \(\tan A\), which can often be related to \(\tan(45^\circ \pm A)\) or \(\tan(A \pm 45^\circ)\).
• For example, \(\frac{\cos A - \sin A}{\cos A + \sin A} = \frac{1 - \tan A}{1 + \tan A} = \tan(45^\circ - A)\).
• Similarly, \(\frac{\cos A + \sin A}{\cos A - \sin A} = \frac{1 + \tan A}{1 - \tan A} = \tan(45^\circ + A)\).

Understanding these patterns helps in quickly simplifying such forms in trigonometric equations.