Problem 150 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(\cos 20^{\circ}=k\) and \(\cos x=2 k^{2}-1\), then find the possible values of \(x\) between \(0^{\circ}\) and \(360^{\circ}\)

Step-by-Step Solution

Answer

The possible values of x between \(0^{\circ}\) and \(360^{\circ}\) are \(40^{\circ}\) and \(320^{\circ}\).

Step 1: Identify the double angle identity

The double angle formula for cosine is \(\cos 2\theta = 2 \cos^2 \theta -1 \). Comparing this to the original equation, we can conclude that \(x = 2\theta\).

Step 2: Substitute value of k

Replacing k with \(\cos 20^{\circ}\) and \(\theta\) with \(20^{\circ}\) in the relation \(x = 2\theta\), we get \(x = 2*20^{\circ} = 40^{\circ}\).

Step 3: Calculate possible values of x

This is the solution for x in the interval \(0^{\circ} - 180^{\circ}\). However, the exercise asks for possible values in the interval \(0^{\circ} - 360^{\circ}\). Because cosine is positive in the first and fourth quadrants, \(x\) can be either \(40^{\circ}\) or \(360^{\circ} - 40^{\circ} = 320^{\circ}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate the six trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) to one another. These identities can be used to simplify complex trigonometric expressions, or to convert one function into another to solve trigonometric equations.

The double angle formulas are a set of trigonometric identities that provide relationships for the sine, cosine, and tangent of double angles. The most relevant for our exercise is the cosine double angle identity, which is given by \[ \cos(2\theta) = 2\cos^2(\theta) - 1 \.\] By knowing a value for \(\cos(\theta)\), you can use this identity to find the value of \(\cos(2\theta)\). Understanding these identities is crucial as they often serve as the bridge between given information and the solution to a problem, as seen in our example where the cosine double angle formula helped us find the possible values for \(x\).

Solving Trigonometric Equations

Solving trigonometric equations requires knowledge of trigonometric identities and how to apply them. The steps involved usually include isolating the trigonometric function in question, using identities to simplify the equation, and then finding the angles that satisfy the equation within a given interval.

For example, the exercise provided requires us to solve for \(x\) using the established relationship \(\cos x=2 k^{2}-1\). By recognizing that this is a cosine double angle identity, we found that \(x = 2\theta\), and by substituting \(\theta\) with \(20^\circ\), we were able to find one value of \(x\). However, to complete the task, we must consider the function's periodicity and the range of possible angles. Cosine, in particular, has a period of \(360^\circ\), and because it repeats its values after each period, we must consider all possible angles within the specified range that will satisfy the equation. This often includes using these angles as reference angles to find related solutions in other quadrants of the unit circle.

Cosine Function Properties

The cosine function properties are essential for finding the values of angles as seen in the exercise we are considering. Some key properties of the cosine function include its periodicity, sign in each quadrant, and symmetry.

• \textbf{Periodicity:} The cosine function has a period of \(2\pi\) radians or \(360^\circ\), meaning the function repeats values every \(360^\circ\).
• \textbf{Sign:} Cosine is positive in the first and fourth quadrants and negative in the second and third quadrants. This is due to the function's correspondence to the x-coordinate of a point on the unit circle.
• \textbf{Symmetry:} Cosine is an even function, which means that \(\cos(-\theta) = \cos(\theta)\), reflecting symmetry about the y-axis.

In the given problem, understanding that the cosine function is positive in the first and fourth quadrants allowed us to determine that the possible values of \(x\) within the \(0^\circ\) to \(360^\circ\) range are \(40^\circ\) and \(320^\circ\). These properties are fundamental when working with trigonometric equations as they guide us to the correct solutions within a given domain.