The term π/3 refers to the angle of 60 degrees, which is one of the fundamental angles in trigonometry. It is an important value that arises in various contexts, particularly in the study of inverse trigonometric functions.
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The angle π/3 is equal to 60 degrees, which is a special angle in the unit circle.
The trigonometric functions of π/3 are: $\sin(\pi/3) = \frac{\sqrt{3}}{2}$, $\cos(\pi/3) = \frac{1}{2}$, and $\tan(\pi/3) = \sqrt{3}$.
The inverse trigonometric functions of π/3 are: $\sin^{-1}(\frac{\sqrt{3}}{2}) = \pi/3$, $\cos^{-1}(\frac{1}{2}) = \pi/3$, and $\tan^{-1}(\sqrt{3}) = \pi/3$.
The angle π/3 is one of the special right triangles in the unit circle, with side lengths of 1, $\sqrt{3}$, and 2.
The value of π/3 is often used in the study of periodic functions, such as the sine and cosine functions.
Review Questions
Explain how the angle π/3 is represented on the unit circle and describe its significance in trigonometry.
The angle π/3, or 60 degrees, is a special angle on the unit circle. It is located at the 30-degree mark, and its trigonometric function values are easy to remember: $\sin(\pi/3) = \frac{\sqrt{3}}{2}$, $\cos(\pi/3) = \frac{1}{2}$, and $\tan(\pi/3) = \sqrt{3}$. This angle is important because it is one of the fundamental angles used in the study of trigonometric functions and their inverse functions. Understanding the properties of π/3 can help students simplify trigonometric expressions and solve a variety of problems involving inverse trigonometric functions.
Describe how the inverse trigonometric functions relate to the angle π/3 and explain their significance in the context of 6.3 Inverse Trigonometric Functions.
The inverse trigonometric functions, such as $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$, are used to find the angle given the value of a trigonometric function. In the case of π/3, the inverse trigonometric functions are: $\sin^{-1}(\frac{\sqrt{3}}{2}) = \pi/3$, $\cos^{-1}(\frac{1}{2}) = \pi/3$, and $\tan^{-1}(\sqrt{3}) = \pi/3$. These inverse functions are crucial in the study of 6.3 Inverse Trigonometric Functions, as they allow students to determine the angle given the value of a trigonometric function, which is a fundamental skill in solving a variety of trigonometric problems.
Analyze how the properties of the angle π/3 can be used to derive trigonometric identities and simplify expressions involving inverse trigonometric functions.
The angle π/3 has several unique properties that can be used to derive important trigonometric identities and simplify expressions involving inverse trigonometric functions. For example, the fact that $\sin(\pi/3) = \frac{\sqrt{3}}{2}$ and $\cos(\pi/3) = \frac{1}{2}$ can be used to establish the identity $\sin^2(\theta) + \cos^2(\theta) = 1$. Additionally, the inverse trigonometric functions of π/3, such as $\sin^{-1}(\frac{\sqrt{3}}{2}) = \pi/3$ and $\tan^{-1}(\sqrt{3}) = \pi/3$, can be used to simplify expressions and solve problems involving inverse trigonometric functions. Understanding the properties of π/3 and how to apply them in various contexts is crucial for success in the study of 6.3 Inverse Trigonometric Functions.