Question

Without using a calculator, evaluate or simplify the following expressions. $$\tan ^{-1} \sqrt{3}$$

Short answer

Answer: The inverse tangent of the square root of 3 without using a calculator is 60°.

Step-by-step solution

Understand Inverse Tangent Function

The inverse tangent function, denoted as \(\tan ^{-1}(x),\) asks for the angle whose tangent value is x. In other words, if \(\tan(\theta) = x,\) then \(\tan^{-1}(x) = \theta.\) Note that for the inverse tangent function, the range of possible angles is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) or between -90° and 90°.

Identify Special Triangles

In order to find the arctangent without a calculator, we want to think about special right triangles: the 30-60-90 triangle and the 45-45-90 triangle. These triangles have specific properties and tangent values for their angles that allow us to easily solve this problem. For a 30-60-90 triangle, the sides are in a ratio of \(1:\sqrt{3}:2.\) The tangent values are: - In the case of the 30° angle (which is \(\frac{\pi}{6}\) in radians), \(\tan(30°) = \frac{1}{\sqrt{3}}\) - In the case of the 60° angle (which is \(\frac{\pi}{3}\) in radians), \(\tan(60°) = \sqrt{3}\) For a 45-45-90 triangle, the sides are in a ratio of \(1:1:\sqrt{2}.\) The tangent values are: - In the case of the 45° angle (which is \(\frac{\pi}{4}\) in radians), \(\tan(45°) = 1\)

Find the Correct Angle

Now we will try to match our given expression, \(\tan^{-1}(\sqrt{3}),\) with the tangent values of the angles in special triangles that we found in step 2. Recall that we have the following tangent values: - \(\tan(30°) = \frac{1}{\sqrt{3}}\) - \(\tan(45°) = 1\) - \(\tan(60°) = \sqrt{3}\) We see that \(\tan(60°) = \sqrt{3}.\) Therefore, we can say that \(\tan^{-1}(\sqrt{3}) = 60°.\) So, the evaluation of the given expression is: $$\tan^{-1}(\sqrt{3}) = 60^{\circ}$$

Key concepts

Special Triangles

Special triangles are a fundamental tool in trigonometry, often providing a shortcut to determine trigonometric ratios like sine, cosine, and tangent without the use of a calculator. There are two important types of special triangles, each with its own set of side ratios and angle measures: the 30-60-90 triangle and the 45-45-90 triangle.

For a 30-60-90 triangle, the sides are in a specific ratio of 1 to \(\sqrt{3}\) to 2. This means:

• The shortest side (opposite the 30° angle) is 1.
• The side opposite the 60° angle is \(\sqrt{3}\).
• And the hypotenuse, opposite the 90° angle, is 2.

On the other hand, the 45-45-90 triangle, which is an isosceles right triangle, has sides in the ratio of 1 to 1 to \(\sqrt{2}\), where both legs are equal, and the hypotenuse is \(\sqrt{2}\).

Recognizing these triangles and their side ratios allows one to quickly calculate tangent values for the angles 30°, 45°, and 60°, which are commonly found in trigonometry problems.

Tangent Function

The tangent function is one of the basic trigonometric functions, which relates the angle of a right triangle to the ratio of the opposite side to the adjacent side. Mathematically, it is expressed as \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

In the context of inverse trigonometric functions, when we see \(\tan^{-1}(x)\), we are essentially seeking the angle \(\theta\) such that \(\tan(\theta) = x\). This makes the inverse tangent function vital for finding angles when given tangent ratios.

The inverse tangent function, or \(\arctan(x)\), has a principal value range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), which corresponds to angles between -90° and 90°. Understanding this range is crucial, as it constrains the possible angles we identify when evaluating expressions like \(\tan^{-1}(\sqrt{3})\). Knowing this helps us find the correct angle within the primary range for the tangent function.

Angle Evaluation

Angle evaluation is the process of determining the measure of an angle that satisfies a given trigonometric equation. In the context of the problem \(\tan^{-1}(\sqrt{3})\), we must find an angle whose tangent is \(\sqrt{3}\). By referring to our knowledge of special triangles, we know this value matches the angle in the 30-60-90 triangle.

From our analysis of special triangles, we remember:

• \(\tan(60°) = \sqrt{3}\)

Since 60° is within the range of \(\tan^{-1}\), which is from -90° to 90°, we can confidently say that \(\tan^{-1}(\sqrt{3}) = 60°\).

Evaluating angles without calculators often involves using this foundational understanding of special triangles and their intrinsic properties. Both the recognition of these unique triangles and the tangent values of their angles are key to solving such inverse trigonometric problems efficiently.