Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find each value of \(\theta\) in degrees \((0^{\circ}<\theta<90^{\circ})\) and radians \((0 < \theta <\pi / 2)\) without using a calculator. (a) \(\tan \theta=\sqrt{3}\) (b) \(\cos \theta=\frac{\sqrt{2}}{2}\)

Step-by-Step Solution

Answer

The value of \(\theta\) for \(\tan \theta =\sqrt{3}\) is \(60^{\circ}\) or \(\frac{\pi}{3}\) and for \(\cos \theta =\frac{\sqrt{2}}{2}\) is \(45^{\circ}\) or \(\frac{\pi}{4}\).

Step 1: Solve for \(\theta\) in \(\tan \theta =\sqrt{3}\)

The angle \(\theta\) that produces a tangent of \(\sqrt{3}\) in the first quadrant is known from the unit circle or special right triangles (30-60-90). This angle in degrees is \(60^{\circ}\) and in radians, it's \(\frac{\pi}{3}\).

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

Using the unit circle or special right triangles (45-45-90), the angle \(\theta\) that yields a cosine of \(\frac{\sqrt{2}}{2}\) in the first quadrant is established. In degrees, the angle \(\theta\) is \(45^{\circ}\) and in radians, it's \(\frac{\pi}{4}\).

Key Concepts

Tangent function

The tangent function, often written as \( \tan \theta \), represents the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. It is a fundamental function in trigonometry that describes certain angular relationships. The tangent function is defined as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)
• Or, on the unit circle, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This function is periodic with a period of \( \pi \), meaning that it repeats its values every \( \pi \) radians. For example, if you know the value of \( \tan 30^{\circ} \), you also know what it will be at \( 210^{\circ} \) since \( 210^{\circ} = 30^{\circ} + 180^{\circ} \).
In practical terms, \( \tan \theta = \sqrt{3} \) corresponds to an angle of \( 60^{\circ} \) or \( \frac{\pi}{3} \) radians. This results from specific angles where sine and cosine have known values, such as in 30-60-90 triangles.

Cosine function

The cosine function, denoted as \( \cos \theta \), provides the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. It is a crucial aspect of trigonometry, aiding in the calculation of angles and other side lengths. Its formula is written as:

• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• And on the unit circle, \( \cos \theta \) is simply the x-coordinate of the point where the terminal side of an angle \( \theta \) intersects the circle.

The cosine function is periodic with a period of \( 2\pi \) radians, which means it completes one full cycle every \( 360^{\circ} \). For the given condition, \( \cos \theta = \frac{\sqrt{2}}{2} \), the angle \( \theta \) in the first quadrant is \( 45^{\circ} \) or \( \frac{\pi}{4} \) radians.
Special right triangles, like 45-45-90 triangles, make this a well-memorized value because the legs of such triangles are equal, leading to simple trigonometric ratios for both sine and cosine.

Unit circle

The unit circle is an essential tool in trigonometry, representing a circle with a radius of one centered at the origin of a coordinate plane. It connects trigonometric functions to geometry and provides a convenient visualization of angle measures in both radians and degrees.
In the unit circle:

• The x-coordinate of any point is \( \cos \theta \).
• The y-coordinate is \( \sin \theta \).
• The radius (always 1) ensures that \( \sin \theta^2 + \cos \theta^2 = 1 \).

For angles like \( 60^{\circ} \) and \( 45^{\circ} \), the points on the unit circle correspond to known sine and cosine values:

• For \( 60^{\circ} \) or \( \frac{\pi}{3} \) radians, the coordinates are \( (\frac{1}{2}, \sqrt{3}/2) \).
• For \( 45^{\circ} \) or \( \frac{\pi}{4} \) radians, the coordinates are \( (\sqrt{2}/2, \sqrt{2}/2) \).

The unit circle simplifies finding these values without calculations and helps visualize the relationships between the trigonometric functions.

Radian measure

Radian measure is a natural and widely used method of measuring angles, particularly in trigonometry and calculus. Contrary to degrees, which divide a circle into \(360\) equal parts, radians are based on the radius of the circle. This makes it extremely useful in mathematical applications.
One radian is the angle created when the arc length equals the radius of the circle. A full circle encompasses \(2\pi\) radians, establishing the essential conversion:

• \(180^{\circ} = \pi \text{ radians} \)
• \(90^{\circ} = \frac{\pi}{2} \text{ radians} \)

This conversion is vital for translating problems from degree to radian measure. For example, the exercise involved finding angles in both degrees and radians, such as \( 60^{\circ} = \frac{\pi}{3} \) and \( 45^{\circ} = \frac{\pi}{4} \). This ability to switch seamlessly between these two systems is an important skill for solving various mathematical problems across disciplines.
By understanding radian measure, one gains more profound insights and flexibilities in applications involving circular motion or periodic functions.