Solving the Trigonometry Equation for sin θ

The problem asks us to find the value of sin θ given a specific equation involving sec θ and tan θ.

The given equation is:

\(\frac {\sec θ + \tan θ }{\sec θ - \tan θ } = 2 \frac {51}{79}\)

First, let's convert the mixed fraction on the right-hand side into an improper fraction:

\(2 \frac {51}{79} = 2 \times \frac{79}{79} + \frac{51}{79} = \frac{158}{79} + \frac{51}{79} = \frac{158 + 51}{79} = \frac{209}{79}\)

So the equation becomes:

\(\frac {\sec θ + \tan θ }{\sec θ - \tan θ } = \frac {209}{79}\)

This equation is in a form where we can conveniently apply the Componendo and Dividendo rule. The rule states that if \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{a+b}{a-b} = \frac{c+d}{c-d}\).

Applying Componendo and Dividendo to our equation:

Let \(a = \sec θ + \tan θ\), \(b = \sec θ - \tan θ\), \(c = 209\), and \(d = 79\).

\(\frac {(\sec θ + \tan θ) + (\sec θ - \tan θ) }{(\sec θ + \tan θ) - (\sec θ - \tan θ) } = \frac {209 + 79}{209 - 79}\)

Simplify the numerator and the denominator on the left side:

• Numerator: \((\sec θ + \tan θ) + (\sec θ - \tan θ) = \sec θ + \tan θ + \sec θ - \tan θ = 2 \sec θ\)
• Denominator: \((\sec θ + \tan θ) - (\sec θ - \tan θ) = \sec θ + \tan θ - \sec θ + \tan θ = 2 \tan θ\)

Simplify the numerator and the denominator on the right side:

• Numerator: \(209 + 79 = 288\)
• Denominator: \(209 - 79 = 130\)

Substituting these simplified expressions back into the equation:

\(\frac {2 \sec θ}{2 \tan θ} = \frac {288}{130}\)

The 2's cancel out on the left side, and we can simplify the fraction on the right side by dividing both numerator and denominator by 2:

\(\frac {\sec θ}{\tan θ} = \frac {144}{65}\)

Now, let's express sec θ and tan θ in terms of sin θ and cos θ. We know that:

• \(\sec θ = \frac{1}{\cos θ}\)
• \(\tan θ = \frac{\sin θ}{\cos θ}\)

Substitute these into the equation \(\frac {\sec θ}{\tan θ} = \frac {144}{65}\):

\(\frac {\frac{1}{\cos θ}}{\frac{\sin θ}{\cos θ}} = \frac {144}{65}\)

Simplify the left side:

\(\frac{1}{\cos θ} \times \frac{\cos θ}{\sin θ} = \frac{1}{\sin θ}\)

So the equation becomes:

\(\frac {1}{\sin θ} = \frac {144}{65}\)

To find sin θ, we take the reciprocal of both sides:

\(\sin θ = \frac {65}{144}\)

This value matches one of the given options.

Comparing with Options

Let's compare our calculated value of sin θ with the given options:

1. \(\frac {35}{72}\)
2. \(\frac {91}{144}\)
3. \(\frac {65}{144}\)
4. \(\frac {39}{72}\)

Our result, \(\sin θ = \frac{65}{144}\), matches Option 3.

Revision Table: Key Trigonometric Identities and Rules

Additional Information on Solving Trigonometry Ratios

When faced with trigonometric equations involving ratios like the one in this problem, applying the Componendo and Dividendo rule is often a very efficient method to simplify the expression. This rule helps transform a ratio of sums and differences into a simpler ratio.

Once simplified, converting the trigonometric functions (like secant, tangent, cosecant, cotangent) back to their fundamental definitions in terms of sine and cosine is a standard approach to solve for sine or cosine, or their reciprocals.

Remember that trigonometric identities are fundamental tools in solving such problems. Knowing the relationships between different trigonometric functions allows you to manipulate equations and isolate the desired variable.

In this specific problem, the steps involved were:

1. Simplifying the numerical part of the equation.
2. Recognizing the structure suitable for Componendo and Dividendo.
3. Applying Componendo and Dividendo to simplify the trigonometric expression.
4. Converting the simplified expression into terms of sine and cosine.
5. Solving for the required trigonometric function, sin θ.

Practicing similar problems will help you quickly identify when and how to apply rules like Componendo and Dividendo and effectively use trigonometric identities.