Problem 45 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \cos \theta-\sqrt{\cos \theta}=0 $$

Step-by-Step Solution

Answer

The solutions are \( \theta = 90^\circ + 180^\circ k \) and \( \theta = 0^\circ + 360^\circ k \), for integers \( k \).

Step 1: Factor the Equation

We start with the equation \( \cos \theta - \sqrt{\cos \theta} = 0 \). We can factor this equation as \( \sqrt{\cos \theta} (\sqrt{\cos \theta} - 1) = 0 \). This implies that either \( \sqrt{\cos \theta} = 0 \) or \( \sqrt{\cos \theta} - 1 = 0 \).

Step 2: Solve \( \sqrt{\cos \theta} = 0 \)

If \( \sqrt{\cos \theta} = 0 \), then \( \cos \theta = 0 \). The solutions for \( \theta \) are angles where the cosine function is zero, which are at \( \theta = 90^\circ + 180^\circ k \), where \( k \) is an integer.

Step 3: Solve \( \sqrt{\cos \theta} - 1 = 0 \)

If \( \sqrt{\cos \theta} - 1 = 0 \), then \( \sqrt{\cos \theta} = 1 \). Squaring both sides gives \( \cos \theta = 1 \). The solution for \( \theta \) is \( \theta = 0^\circ + 360^\circ k \), where \( k \) is an integer.

Step 4: Combine the Solutions

The solutions are \( \theta = 90^\circ + 180^\circ k \) from solving \( \cos \theta = 0 \) and \( \theta = 0^\circ + 360^\circ k \) from solving \( \cos \theta = 1 \). Both sets of solutions need to be considered as \( k \) varies over all integers.

Key Concepts

Factorization

Factorization is a crucial algebraic process that simplifies complex expressions into products of simpler terms, making equations more manageable. In trigonometric equations, like the one given, factorization helps in breaking down expressions involving trigonometric functions. The equation \( \cos \theta - \sqrt{\cos \theta} = 0 \) can be factored as \( \sqrt{\cos \theta} (\sqrt{\cos \theta} - 1) = 0 \), dividing it into simpler parts.

• By setting the equation to zero, the product rule tells us that either term in the factorization must be zero.
• This leads to two separate conditions: \( \sqrt{\cos \theta} = 0 \) and \( \sqrt{\cos \theta} - 1 = 0 \).

Understanding how trigonometric functions can be factored allows for more straightforward solutions by identifying the core terms that must each independently satisfy the equation.

Angle Measurement in Degrees

Angle measurement in degrees is a way to quantify the size of an angle, crucial in solving trigonometric equations that are based on the unit circle. One complete rotation around a circle is \(360^\circ\). Trigonometric functions often rely on angle measurements to define their values at various points around the circle.

• The angle at which the cosine of \( \theta \) is \(0\) are \(90^\circ + 180^\circ k\), where \(k\) is an integer.
• The angle at which the cosine is \(1\) is \(0^\circ + 360^\circ k\), for integer \(k\).
• These solutions represent the angles on the unit circle where the cosine takes specific values, allowing for multiple solutions in the set of real numbers.

It is crucial to understand how angles cycle through periodic intervals, helping to arrive at all possible solutions for any trigonometric equation defined over the real number set.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and is crucial in solving problems involving cycles and periodicity. It relates an angle to the x-coordinate of a point on the unit circle. For angle \(\theta\), \(\cos \theta\) expresses the horizontal distance from the origin on the unit circle.

• The equation \(\cos \theta = 0\) indicates points where the x-coordinate is zero, occurring at \(90^\circ + 180^\circ k\).
• When \(\cos \theta = 1\), this means the x-coordinate reaches its maximum, located at \(0^\circ + 360^\circ k\).

The periodic nature of the cosine function, with a period of \(360^\circ\), signifies that solutions repeat after every full rotation. Grasping the cosine function's periodicity helps understand the multi-solution nature of trigonometric equations across different angle measurements.