Problem 58 - Precalculus: A Concise Course

Question

Find the values of \(\theta\) in degrees \(\left(0^{\circ}<\theta<90^{\circ}\right)\) and radians \((0<\theta<\pi / 2)\) without the aid of a calculator. (a) \(\cos \theta=\frac{\sqrt{2}}{2}\) (b) \(\tan \theta=1\)

Step-by-Step Solution

Answer

The answer to part (a) is \(\theta = 45^\circ\) or \(\theta = \frac{\pi}{4}\) radians and to part (b) is \(\theta = 45^\circ\) or \(\theta = \frac{\pi}{4}\) radians.

Step 1: Identify special angles and corresponding trigonometric ratios

We know from the unit circle that some angles have distinctive cosine and tangent values. Sepecifically, for angles \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\) (or \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\) in radians), the cosine and tangent values are well known.

Step 2: Find the value of \(\theta\) satisfying \(\cos \theta=\frac{\sqrt{2}}{2}\)

By referring to standard values, it's recognized that cosine value of \(\frac{\sqrt{2}}{2}\) corresponds to an angle of \(45^\circ\) or \(\frac{\pi}{4}\) radians. Hence, the solution for the part (a) is \(\theta = 45^\circ\) or \(\theta = \frac{\pi}{4}\) radians.

Step 3: Find the value of \(\theta\) satisfying \(\tan \theta = 1\)

By referring to standard values, it's recognized that tangent value of \(1\) corresponds to an angle of \(45^\circ\) or \(\frac{\pi}{4}\) radians. Hence, the solution for the part (b) is \(\theta = 45^\circ\) or \(\theta = \frac{\pi}{4}\) radians.

Key Concepts

Special Angles

Special angles are used frequently in trigonometry because of their easy-to-remember trigonometric values. These angles include common measures like

• 0°
• 30°
• 45°
• 60°
• 90°

These are equivalent in radians to:

• 0
• \(\frac{\pi}{6}\)
• \(\frac{\pi}{4}\)
• \(\frac{\pi}{3}\)
• \(\frac{\pi}{2}\)

Special angles are helpful because their sine, cosine, and tangent ratios are often memorized. This makes solving trigonometric problems straightforward, without a calculator. For example, knowing that the cosine of 45° (or \(\frac{\pi}{4}\)) is \(\frac{\sqrt{2}}{2}\), can quickly lead you to the solution in exercises involving this value.

Cosine Function

The cosine function is one of the primary functions in trigonometry. It relates the angle in a right triangle to the ratio of the adjacent side over the hypotenuse. In the context of the unit circle, the cosine function represents the x-coordinate of a point on the circle as it relates to the angle made with the positive x-axis.

Understanding the unit circle helps in determining the cosine values for special angles. For example, for 45° or \(\frac{\pi}{4}\), the cosine value is \(\frac{\sqrt{2}}{2}\). This value arises because, in an isosceles right triangle (45-45-90 triangle), both legs are equal, making the hypotenuse's length \(\sqrt{2}\) times the leg length.

Tangent Function

The tangent function relates to the angle in a right triangle as the ratio of the opposite side over the adjacent side. On the unit circle, the tangent of an angle is the y-coordinate divided by the x-coordinate when the point lies on the circumference.

The tangent of 45° or \(\frac{\pi}{4}\) is simple and known to be 1. This comes from the fact that, in a 45-45-90 triangle, the opposite and adjacent sides are equal, so their ratio is 1. This specific tangent value, like others for special angles, is crucial in quickly solving basic trigonometric equations.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate system. It is an indispensable tool for exploring trigonometric functions. The circle helps visualize the sine, cosine, and tangent functions as it relates angles to coordinates and ratios.

Each point on the unit circle corresponds to an angle and provides an easy way to see the relationship between angles and trigonometric ratios. For instance, the coordinates of the point on the circle at a 45° angle (or \(\frac{\pi}{4}\) radians) are \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\). Because the y-coordinate equals the x-coordinate at this point, the tangent is 1, and the cosine is \(\frac{\sqrt{2}}{2}\).

The unit circle remains an important concept for quickly understanding and calculating trigonometric values for angles without needing additional tools like calculators.