Problem 108 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find two solutions of each equation. Give your solutions in both degrees \(\left(0^{\circ} \leq \theta < 360^{\circ}\right)\) and radians \((0 \leq \theta < 2 \pi) .\) Do not use a calculator. (a) \(\cos \theta=\frac{\sqrt{2}}{2}\) (b) \(\cos \theta=-\frac{\sqrt{2}}{2}\)

Step-by-Step Solution

Answer

The solutions of the equation \(\cos\theta=\frac{\sqrt{2}}{2}\) are 45 and 315 degrees or \(\frac{\pi}{4}\) and \(\frac{7\pi}{4}\) radians. The solutions of the equations \(\cos\theta=-\frac{\sqrt{2}}{2}\) are 135 and 225 degrees or \(\frac{3\pi}{4}\) and \(\frac{5\pi}{4}\) radians.

Step 1: Solve \(\cos\theta=\frac{\sqrt{2}}{2}\) in degrees

In this case, by referring to the values of cosine function in the unit circle, \(\theta\) equals 45 degrees and 315 degrees in the interval \(\[0^{\circ}, 360^{\circ})\). It's because the cosine of these angles gives a value of \(\frac{\sqrt{2}}{2}\).

Step 2: Convert the degrees to radians for the first equation

To convert 45 degrees to radians, multiply by \(\frac{\pi}{180}\) to get \(\frac{\pi}{4}\) radians. Similarly, 315 degrees in radians is \(\frac{7\pi}{4}\). So, the solutions of \(\cos\theta=\frac{\sqrt{2}}{2}\) in radians are \(\frac{\pi}{4}\) and \(\frac{7\pi}{4}\).

Step 3: Solve \(\cos\theta=-\frac{\sqrt{2}}{2}\) in degrees

By observing the unit circle, \(\theta\) equals 135 degrees and 225 degrees in the interval \(0^{\circ}, 360^{\circ})\) because the cosine of these angles gives a value of \(-\frac{\sqrt{2}}{2}\).

Step 4: Convert the degrees to radians for the second equation

135 degrees equal \(\frac{3\pi}{4}\) radians and 225 degrees equal \(\frac{5\pi}{4}\) radians. Hence, the solutions of \(\cos\theta=-\frac{\sqrt{2}}{2}\) in radians are \(\frac{3\pi}{4}\) and \(\frac{5\pi}{4}\)

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in the study of triangles, circles, and oscillatory motion. They are often represented as ratios of the sides of a right triangle relative to an angle within the triangle. Among these functions, the cosine graph plays a pivotal role in capturing the horizontal component of a point on the unit circle.

For instance, when solving the equation \(\cos \theta = \frac{\sqrt{2}}{2}\), what we're actually trying to find are the angles at which the horizontal distance from the origin to a point on the unit circle's circumference is \frac{\sqrt{2}}{2}. Likewise, when \(\cos \theta = -\frac{\sqrt{2}}{2}\), we seek the angles where this horizontal distance is negative, indicating that the point lies on the left half of the unit circle. Trigonometric equations like these often have two solutions within one full rotation because the unit circle is symmetrical about the x-axis.

Unit Circle

The unit circle is a powerful tool for understanding trigonometric functions. With a radius of 1 unit, it's centered at the origin of a coordinate plane. Each point on the periphery corresponds to an angle measure, from the positive x-axis, and provides values for sine and cosine.

In the case of our exercise, the cosine value is given, and we must find the corresponding angles on the unit circle. The provided values \(\frac{\sqrt{2}}{2}\) and \(-\frac{\sqrt{2}}{2}\) are particularly useful because they correspond to angles that form 45-degree increments. These angles are commonly found at the intersections of the unit circle with lines that divide the circle into eight equal segments.

Radians and Degrees Conversion

Radians and degrees are two units for measuring angles, and converting between them is crucial in trigonometry. Degrees are often the first unit students learn; they divide a circle into 360 equal parts. On the other hand, radians provide a different perspective: they relate an angle to the radius of the circle, with one radian being the measure of an angle at the center of a circle that intercepts an arc equal in length to the radius.

Converting from degrees to radians involves the mathematical constant \(\pi\), which is approximately 3.14159. To convert degrees to radians, one multiplies by \(\frac{\pi}{180}\), since \(180^\circ\) corresponds to \(\pi\) radians. Students can use this relationship to express the solutions of trigonometric equations in both degrees and radians, allowing for a deeper understanding and versatility in mathematics.

Cosine Values

Cosine values are closely tied to the x-coordinate of a point on the unit circle. They have a range from -1 to 1 and are key in solving trigonometric equations. In our exercise, we encounter cosine values of \(\frac{\sqrt{2}}{2}\) and \(-\frac{\sqrt{2}}{2}\), which are notable because they correspond to angles where a right triangle with equal legs is inscribed in the unit circle.

In the context of these special angles, such as 45, 135, 225, and 315 degrees—or in radians, \(\pi/4\), \(3\pi/4\), \(5\pi/4\), and \(7\pi/4\)—we can find exact cosine values without a calculator. Recognizing and memorizing these common cosine values helps students to solve trigonometric equations more efficiently and deepen their comprehension of the cyclical nature of trigonometric functions.