Question

If tan \(\theta=\sqrt{2}\), then the value of \(\theta\) is : (a) less than \(\frac{\pi}{4}\) (b) equal to \(\frac{\pi}{4}\) (c) between \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\) (d) greater than \(\frac{\pi}{3}\)

Short answer

Answer: (c) between 𝜋/4 and 𝜋/3

Step-by-step solution

Recall the definition of the tangent function

The tangent function (tan) is defined as the ratio of the length of the opposite side to the length of the adjacent side of a right triangle. In other words, \(\tan{\theta}= \frac{opposite}{adjacent}\).

Recognize the given information

The problem provides the value of the tangent function as \(\tan{\theta} = \sqrt{2}\).

Analyze the properties of angles in the right triangle

In a right triangle, the opposite side grows larger while the adjacent side gets smaller as the angle \(\theta\) increases from 0 to \(\frac{\pi}{2}\). Consequently, the value of tan grows from 0 to infinity in the same range. Meanwhile, the angles of \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\) play significant roles in trigonometry, with tan values at these angles having specific values: 1. At \(\theta = \frac{\pi}{4}\), \(\tan{\theta} = 1\). 2. At \(\theta = \frac{\pi}{3}\), \(\tan{\theta} = \sqrt{3} \approx 1.73\).

Compare the value of the tangent function

By comparing the given value of the tangent function (\(\tan{\theta} = \sqrt{2} \approx 1.41\)) with the values at \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\), we can determine the possible range of values for \(\theta\): 1. \(\tan{\theta} = \sqrt{2} > 1\). 2. \(\tan{\theta} = \sqrt{2} < \sqrt{3}\).

Determine the range of angle \(\theta\)

Since \(\tan{\theta} > \tan{\frac{\pi}{4}}\) and \(\tan{\theta} < \tan{\frac{\pi}{3}}\), \(\theta\) must be between \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\). Thus, the answer is (c) between \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\).

Key concepts

Trigonometric Ratios

Getting to know trigonometric ratios is key to understanding various properties of right triangles and the relationships between their angles and sides. In simple terms, trigonometric ratios provide a numerical relationship that correlate the angles of a right triangle to the lengths of its sides.

For instance, when we look at the tangent function, it's one of the primary trigonometric functions, often abbreviated to 'tan'. The tangent of an angle within a right triangle is the ratio between the length of the side opposite to the angle and the length of the adjacent side (the one next to the angle but not the hypotenuse). Notably, this ratio changes as the angle increases or decreases, but the relationship is always constant for a given angle, regardless of the size of the triangle.

Understanding this concept is fundamental, as it can be applied to solve various geometric problems, such as determining the slope of a line, calculating the height of an object using shadow length, and navigating by triangulation.

Right Triangle Properties

Right triangles have properties that set them apart from other types of triangles. The most distinctive feature is the right angle, which is exactly 90 degrees or \(\frac{\pi}{2}\) radians. This right angle creates a unique relationship between the sides, defined by the Pythagorean theorem as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, the longest side of the triangle opposite the right angle, and \(a\) and \(b\) are the other two sides.

In the context of trigonometry, we use specific names for the sides in relation to a certain angle (other than the right angle), typically labeled as \(\theta\). The side 'adjacent' to \(\theta\) is next to it, the side 'opposite' is across from it, and the 'hypotenuse' remains the longest side. The values of trigonometric functions like sine, cosine, and tangent are determined based on these sides' lengths. For example, in the provided exercise, understanding that the tangent function is a ratio of the 'opposite' to the 'adjacent' is crucial.

Angle Measurement in Radians

Angle measurement is a substantial concept in trigonometry, with radians being one of the two main units of measurement, the other being degrees. Radians offer a way of measuring angles based on the radius of a circle. A full circle is \(2\pi\) radians, which is equivalent to 360 degrees. This means that one radian is approximately 57.2958 degrees.

An angle measured in radians provides a direct correlation to the arc length of a circle, with one radian being the angle subtended by an arc that is equal in length to the radius of the circle. This relationship is particularly useful in calculus and other areas of advanced mathematics. For example, the common angles of \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\) radians correspond to 45 and 60 degrees, respectively. In the exercise solution provided, we can see the practical application of radians; given the tangent value, we used radians as the measurement to determine between which two standard angles the unknown angle \(\theta\) lies.