Chapter 4.2, Problem 27E

To determine

Find the real number values of Sine, Cosine and Tangent of the angle $\text{t}\text{=}\frac{\text{2π}}{\text{3}}$

Answer to Problem 27E

  $\text{sin}\frac{\text{2π}}{\text{3}}=\frac{\sqrt{3}}{\text{2}};\text{cos}\frac{\text{2π}}{\text{3}}=-\frac{1}{\text{2}};\text{tan}\frac{\text{2π}}{\text{3}}=-\sqrt{3}$

Explanation of Solution

Given: $\text{t}\text{=}\frac{\text{2π}}{\text{3}}$

Concept Used:

For each t − value, begin by finding the corresponding point (x,y) on the unit circle. Then use the definition of trigonometric Functions.

For unit circle, the radius = 1

We know that in Trigonometry:

  $\text{sinθ}\text{=}y;\text{cosθ}\text{=}x;\text{tanθ}\text{=}\frac{\text{sinθ}}{\text{cosθ}}=\frac{\text{y}}{\text{x}}$

Calculation:

  $\text{sin}\frac{\text{2π}}{\text{3}}\text{=}y;\text{cos}\frac{\text{2π}}{\text{3}}\text{=}x;\text{tan}\frac{\text{2π}}{\text{3}}\text{=}\frac{\text{sin}\frac{\text{2π}}{\text{3}}}{\text{cos}\frac{\text{2π}}{\text{3}}}=\frac{\text{y}}{\text{x}};$

From the unit circle, $x=-\frac{1}{\text{2}};y=\frac{\sqrt{3}}{\text{2}}$

  $\text{sin}\frac{\text{2π}}{\text{3}}\text{=}y=\frac{\sqrt{3}}{\text{2}};\text{cos}\frac{\text{2π}}{\text{3}}\text{=}x=-\frac{1}{\text{2}};\text{tan}\frac{\text{2π}}{\text{3}}\text{=}\frac{\text{sin}\frac{\text{2π}}{\text{3}}}{\text{cos}\frac{\text{2π}}{\text{3}}}=\frac{\raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=-\sqrt{3}$

Thus, Sine, Cosine and Tangent Trigonometric functions of the real number for $\text{t}\text{=}\frac{\text{2π}}{\text{3}}$ are:

  $\text{sin}\frac{\text{2π}}{\text{3}}=\frac{\sqrt{3}}{\text{2}};\text{cos}\frac{\text{2π}}{\text{3}}=-\frac{1}{\text{2}};\text{tan}\frac{\text{2π}}{\text{3}}=-\sqrt{3}$