Chapter 4.2, Problem 38E

To determine

Find the real number values of Sine, Cosine and Tangent of the angle $\text{t}=\frac{\text{7π}}{\text{4}}$

Answer to Problem 38E

  $\text{sin}\frac{\text{3π}}{\text{4}}=-\frac{\sqrt{2}}{\text{2}};\text{cos}\frac{\text{3π}}{\text{4}}=\frac{\sqrt{2}}{\text{2}};\text{tan}\frac{\text{3π}}{\text{4}}=-1$

Explanation of Solution

Given: $\text{t}=\frac{\text{7π}}{\text{4}}$

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:

  $\text{sin}\frac{\text{7π}}{\text{4}}\text{=}y;\text{cos}\frac{\text{7π}}{\text{4}}\text{=}x;\text{tan}\frac{\text{7π}}{\text{4}}\text{=}\frac{\text{sin}\frac{\text{7π}}{\text{4}}}{\text{cos}\frac{\text{7π}}{\text{4}}}=\frac{\text{y}}{\text{x}}$

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

  $\text{sin}\frac{\text{7π}}{\text{4}}\text{=}y=-\frac{\sqrt{2}}{\text{2}};\text{cos}\frac{\text{7π}}{\text{4}}\text{=}x=\frac{\sqrt{2}}{\text{2}};\text{tan}\frac{\text{7π}}{\text{4}}\text{=}\frac{\text{sin}\frac{\text{7π}}{\text{4}}}{\text{cos}\frac{\text{7π}}{\text{4}}}=-\frac{\raisebox{1ex}{$-\sqrt{2}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\raisebox{1ex}{$\sqrt{2}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=-1;$

Thus, Sine, Cosine and Tangent Trigonometric functions of the real number for $\text{t}=\frac{\text{7π}}{\text{4}}$ are: $\text{sin}\frac{\text{3π}}{\text{4}}=-\frac{\sqrt{2}}{\text{2}};\text{cos}\frac{\text{3π}}{\text{4}}=\frac{\sqrt{2}}{\text{2}};\text{tan}\frac{\text{3π}}{\text{4}}=-1$