Chapter 4.2, Problem 36E

To determine

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

Answer to Problem 36E

  $\text{sin}\frac{\text{3π}}{\text{2}}=-1; \text{cos}\frac{\text{3π}}{\text{2}}=0;\text{tan}\frac{\text{3π}}{\text{2}}=\text{undefined};$

Explanation of Solution

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

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:

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

From the unit circle, $x=0;y=-1$

  $\text{sin}\frac{\text{3π}}{\text{2}}\text{=}y=-1; \text{cos}\frac{\text{3π}}{\text{2}}\text{=}x=0;\text{tan}\frac{\text{3π}}{\text{2}}\text{=}\frac{\text{sin}\frac{\text{3π}}{\text{2}}}{\text{cos}\frac{\text{3π}}{\text{2}}}=\frac{-1}{0}=\text{undefined};$

Thus, Sine, Cosine and Tangent Trigonometric functions of the real number for $\text{t}\text{=}\frac{\text{3π}}{\text{2}}$ are: $\text{sin}\frac{\text{3π}}{\text{2}}=-1; \text{cos}\frac{\text{3π}}{\text{2}}=0;\text{tan}\frac{\text{3π}}{\text{2}}=\text{undefined};$