Solving Trigonometric Equations: sec 4θ = cosec (θ + 20°)

The problem asks us to find the value of the angle θ given the trigonometric equation: \( \sec 4\theta = \operatorname{cosec} (\theta + 20^\circ) \). To solve this equation, we need to use the relationships between trigonometric functions of complementary angles.

Understanding Complementary Angle Identities

Complementary angles are two angles that add up to \(90^\circ\). There are several identities that relate trigonometric functions of complementary angles. A key identity for this problem is the relationship between secant and cosecant:

• \( \sec A = \operatorname{cosec} (90^\circ - A) \)
• \( \operatorname{cosec} A = \sec (90^\circ - A) \)

We can use the first identity, \( \sec A = \operatorname{cosec} (90^\circ - A) \), to rewrite the left side of our given equation, \( \sec 4\theta \).

Applying the Identity and Solving for θ

Given the equation:

\( \sec 4\theta = \operatorname{cosec} (\theta + 20^\circ) \)

Using the identity \( \sec A = \operatorname{cosec} (90^\circ - A) \), we can replace \(A\) with \(4\theta\). So, \( \sec 4\theta = \operatorname{cosec} (90^\circ - 4\theta) \).

Substitute this into the original equation:

\( \operatorname{cosec} (90^\circ - 4\theta) = \operatorname{cosec} (\theta + 20^\circ) \)

If \( \operatorname{cosec} X = \operatorname{cosec} Y \), then in general, \( X = n \cdot 180^\circ + (-1)^n Y \) where \(n\) is an integer. However, for typical problems involving angles in degrees and within common ranges, we consider the case where the angles themselves are equal or related through \(180^\circ\) difference (though for cosecant, \(180^\circ - Y\) is also related). The most straightforward case for acute angles is simply equating the arguments of the cosecant function:

\( 90^\circ - 4\theta = \theta + 20^\circ \)

Now, we can solve this linear equation for θ:

1. Gather terms involving θ on one side and constant terms on the other side.

\( 90^\circ - 20^\circ = \theta + 4\theta \)

2. Simplify both sides.

\( 70^\circ = 5\theta \)

3. Divide by 5 to find the value of θ.

\( \theta = \frac{70^\circ}{5} \)

\( \theta = 14^\circ \)

Verifying the Solution

We found \( \theta = 14^\circ \). Let's check if this value makes the original equation true:

• Left side: \( \sec 4\theta = \sec (4 \times 14^\circ) = \sec 56^\circ \)
• Right side: \( \operatorname{cosec} (\theta + 20^\circ) = \operatorname{cosec} (14^\circ + 20^\circ) = \operatorname{cosec} 34^\circ \)

Using a calculator, \( \sec 56^\circ \approx 1.788 \) and \( \operatorname{cosec} 34^\circ = \frac{1}{\sin 34^\circ} \approx \frac{1}{0.559} \approx 1.788 \). Since \( \sec 56^\circ = \operatorname{cosec} 34^\circ \), and \(56^\circ + 34^\circ = 90^\circ\), our solution \( \theta = 14^\circ \) is correct because \( \sec A = \operatorname{cosec} B \) implies \( A + B = 90^\circ \) (for acute angles A and B).

The value \( \theta = 14^\circ \) is present in the given options.

Revision Table: Key Trigonometric Identities

Additional Information: Solving Trigonometric Equations

Solving trigonometric equations often involves several steps:

1. Simplify the equation: Use identities to express the equation in terms of a single trigonometric function, if possible.
2. Isolate the trigonometric function: Manipulate the equation algebraically to get the trigonometric function (like \(\sin \theta\), \(\cos \theta\), etc.) by itself on one side.
3. Find the reference angle: Determine the angle whose trigonometric function value matches the isolated value.
4. Identify quadrants: Use the sign of the trigonometric value to determine the possible quadrants where the angle could lie.
5. Find the general solution: Write down the general solution considering the periodicity of the trigonometric function and the quadrants identified. For example, if \( \sin \theta = k \), then \( \theta = n\pi + (-1)^n \alpha \), where \( \alpha \) is the principal value. If \( \cos \theta = k \), then \( \theta = 2n\pi \pm \alpha \). If \( \tan \theta = k \), then \( \theta = n\pi + \alpha \). (Note: use degrees or radians consistently).
6. Find particular solutions: If a specific interval for θ is given (e.g., \( 0^\circ \le \theta < 360^\circ \)), find the values of θ within that interval by substituting different integer values for \( n \) in the general solution.

In this specific problem, using the complementary angle identity simplified the equation directly into a linear equation in θ, making steps 3-5 less complex as we assumed the principal relationship between the angles whose cosecant is equal.