Problem 75 - Algebra 2

Question

Which of the following is NOT equal to \(60^{\circ} ?\) $$\begin{array}{llll}{\text { A. } \sin ^{-1} \frac{\sqrt{3}}{2}} & {\text { B. } \cos ^{-1} \frac{1}{2}} & {\text { C. } \tan ^{-1} \sqrt{3}} & {\text { D. } \tan ^{-1} \frac{\sqrt{3}}{3}}\end{array}$$

Step-by-Step Solution

Answer

The answer is D. The inverse tangent of \(\sqrt{3}/3\) is not equal to \(60^{\circ}\), it's \(30^{\circ}\).

Step 1: Calculate the inverse sine

The inverse sine of \(\sqrt{3}/2\) is \(60^{\circ}\), according to the trigonometric table.

Step 2: Calculate the inverse cosine

The inverse cosine of \(1/2\) is \(60^{\circ}\), according to the trigonometric table.

Step 3: Calculate the inverse tangent

The inverse tangent of \(\sqrt{3}\) is \(60^{\circ}\), according to the trigonometric table.

Step 4: Calculate the inverse tangent of another expression

The inverse tangent of \(\sqrt{3}/3\) corresponds to \(30^{\circ}\), not \(60^{\circ}\), according to the trigonometric table. Therefore, this is the choice that is not equivalent to \(60^{\circ}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the tools we use to find the angle when the value of the trigonometric function is known. They are the inverse operations of the basic sine, cosine, and tangent functions. Let's break down these functions:

• Inverse Sine, denoted as \( \sin^{-1} \), is the function used to find an angle whose sine value is known. For example, \( \sin^{-1}(\frac{\sqrt{3}}{2}) = 60^{\circ} \).
• Inverse Cosine, denoted as \( \cos^{-1} \), finds an angle when the cosine value is known. For instance, \( \cos^{-1}(\frac{1}{2}) = 60^{\circ} \).
• Inverse Tangent, denoted as \( \tan^{-1} \), is used to calculate the angle whose tangent is known, such as \( \tan^{-1}(\sqrt{3}) = 60^{\circ} \). However, \( \tan^{-1}(\frac{\sqrt{3}}{3}) = 30^{\circ} \), which is an exception in this exercise.

Each inverse trigonometric function has its specific range of results, ensuring that they return a single value for each input value. This makes them very useful in geometry and calculus.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable involved where both sides are defined. These identities are crucial in simplifying expressions and solving complex equations. Here are some fundamental identities:

• Pythagorean Identity: \( \sin^2\theta + \cos^2\theta = 1 \). This identity relates the squares of the sine and cosine of an angle.
• Reciprocal Identities: Such as \( \csc\theta = \frac{1}{\sin\theta} \), \( \sec\theta = \frac{1}{\cos\theta} \), and \( \cot\theta = \frac{1}{\tan\theta} \).
• Angle Sum and Difference Identities: These describe how to find the sine, cosine, or tangent of the sum or difference of two angles, essential in evaluating complex angles.

Understanding these identities is pivotal. They help to transform complex expressions into more manageable forms and are used extensively in calculus, physics, and engineering.

Trigonometric Angles

In trigonometry, certain angles are so common that they have become special values, referred to as "standard angles." The angles \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\) are among these standard angles, frequently appearing in problems and applications.

• For \(30^{\circ}\), \( \sin 30^{\circ} = \frac{1}{2} \), \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), and \( \tan 30^{\circ} = \frac{\sqrt{3}}{3} \).
• For \(60^{\circ}\), \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), \( \cos 60^{\circ} = \frac{1}{2} \), and \( \tan 60^{\circ} = \sqrt{3} \).
• For \(45^{\circ}\), both sine and cosine are equal: \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), and \( \tan 45^{\circ} = 1 \).

These ratios are ingrained in the study of triangles and circles, making them fundamental tools for solving geometric problems. Recognizing their values can expedite problem-solving and increase accuracy.