Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the expression without using a calculator. $$\tan ^{-1} \frac{\sqrt{3}}{3}$$

Step-by-Step Solution

Answer

\(\tan^{-1}( \frac{\sqrt{3}}{3}) = 30^\circ\)

Step 1: Understanding the Inverse Tangent

The inverse tangent function, \(\tan^{-1}(x)\), is used to find the angle whose tangent is \(x\). The result of this function will give an angle in radians or degrees depending on the convention being used.

Step 2: Evaluate the expression

To find the angle for \(\tan^{-1} (\frac{\sqrt{3}}{3})\), we have to find out for which angle within a standard right angle triangle, the tangent is \(\frac{\sqrt{3}}{3}\). As we know that \(\tan 30^\circ = \frac{\sqrt{3}}{3}\), the result of \(\tan^{-1}( \frac{\sqrt{3}}{3})\) would be \(30^\circ\).

Step 3: Write down the final answer

The angle whose tangent is \(\frac{\sqrt{3}}{3}\) is \(30^\circ\). Therefore, \(\tan^{-1}( \frac{\sqrt{3}}{3}) = 30^\circ\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, used to relate the angles of a triangle to the lengths of its sides. They are especially useful in the study of right triangles and circles. Classical trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each function is associated with a particular ratio of sides within a right triangle.

• The sine function associates an angle with the ratio of the opposite side to the hypotenuse.
• The cosine function relates the adjacent side to the hypotenuse.
• The tangent function is the ratio of the opposite side to the adjacent side.

Inverse trigonometric functions, like \(\tan^{-1}\), are used to find the angle when given these ratios. Understanding these relationships is essential for solving various mathematical problems involving angles and lengths.

Tangent Function

The tangent function, noted as \(\tan(\theta)\), is quite simple yet incredibly useful. It's defined as the ratio of the length of the opposite side to the adjacent side in a right-angled triangle. The formula is:\[\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\]This function is periodic with a period of \(\pi\) radians or \(180^\circ\), meaning it repeats its values over every \(\pi\) radians interval. It is an important function for modeling cyclical behavior.

The inverse tangent function, represented as \(\tan^{-1}(x)\) or \(\arctan(x)\), helps determine the angle \(\theta\) when you know the tangent ratio. For example, if you have \(\tan^{-1}(\frac{\sqrt{3}}{3})\), it tells us the angle where this ratio holds true, which is \(30^\circ\) or \(\frac{\pi}{6}\) radians.

Angle Conversion

Converting angles from degrees to radians and vice versa is an essential skill in trigonometry. While degrees are commonly used in everyday life, radians are more natural in calculus and higher mathematics. The conversion between the two is straightforward:

• To convert degrees to radians, multiply by \(\frac{\pi}{180}\).
• To convert radians to degrees, multiply by \(\frac{180}{\pi}\).

For instance, the angle \(30^\circ\) in degrees converts to radians as follows:\[30^\circ \times \frac{\pi}{180} = \frac{\pi}{6}\text{ radians}\]Understanding these conversions is particularly helpful when dealing with trigonometric functions, as they often require angle values in a specific unit based on the context.

Radians and Degrees

Radians and degrees are two units of measuring angles. Each unit serves a unique purpose in mathematics and science. Degrees, ranging from \(0^\circ\) to \(360^\circ\), are easier for quick, everyday use. They divide the circle into 360 equal parts, which is convenient for navigational purposes.

Radians, on the other hand, are based on the radius of a circle. In radians, a full circle is \(2\pi\). The advantage of radians emerges in calculus, where derivatives and integrals involving trigonometric functions are greatly simplified. For example, \(\pi\) radians is equivalent to \(180^\circ\).

Being comfortable with both units and converting between them is vital in trigonometry, allowing for a more flexible understanding of mathematical concepts.