Chapter 4.2, Problem 88E

(a)

To determine

Find all the Trigonometric Function exists.

Answer to Problem 88E

Explanation of Solution

Given: the angle $\text{θ}$ with coordinate $\text{(x}\text{,}\text{y}\text{)}$

Concept Used:

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. The most widely used trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent,

  

For unit circle: Hypotenuse = Radius of the circle = 1

  

From the unit circle, we know that: $\text{sin}\text{θ}=\text{y};\cos \text{θ}=\text{x};\tan \text{θ}=\frac{\text{sin}\text{θ}}{\text{cos}\text{θ}}=\frac{\text{y}}{\text{x}}$

  $\begin{array}{l}\text{sin}\text{θ}=\text{y};\csc \text{θ}=\frac{1}{\sin \text{θ}}=\frac{1}{\text{y}}\\ \cos \text{θ}=\text{x};\sec \text{θ}=\frac{1}{\cos \text{θ}}=\frac{1}{\text{x}}\\ \tan \text{θ}=\frac{\text{sin}\text{θ}}{\text{cos}\text{θ}}=\frac{\text{y}}{\text{x}};\cot \text{θ}=\frac{1}{\tan \text{θ}}=\frac{x}{\text{y}}\end{array}$

For he angle $\text{θ}$ all six Trigonometric functions exists.

Calculation:

  $\begin{array}{l}\text{sin}\text{θ}=\text{y};\csc \text{θ}=\frac{1}{\sin \text{θ}}=\frac{1}{\text{y}}\\ \cos \text{θ}=\text{x};\sec \text{θ}=\frac{1}{\cos \text{θ}}=\frac{1}{\text{x}}\\ \tan \text{θ}=\frac{\text{sin}\text{θ}}{\text{cos}\text{θ}}=\frac{\text{y}}{\text{x}};\cot \text{θ}=\frac{1}{\tan \text{θ}}=\frac{x}{\text{y}}\end{array}$

Thus, all six trigonometric functions are $\text{sin}\text{θ}=\text{y};\csc \text{θ}=\frac{1}{\text{y}};\cos \text{θ}=\text{x};\sec \text{θ}=\frac{1}{\text{x}};\tan \text{θ}=\frac{\text{y}}{\text{x}};\cot \text{θ}=\frac{x}{\text{y}}$

(b)

To determine

Which Trigonometric functions are positive and negative ?

Answer to Problem 88E

tan $\text{θ}$ and cot $\text{θ}$ are positive only. Sine, cosine, cosecant and secant all are negative.

Explanation of Solution

Given: the angle $\text{θ}$ with coordinate $\text{(x}\text{,}\text{y}\text{)}$

Concept Used:

The angle   $\text{θ}$ in the third quadrant, according to the sign convention rule, tangent and cotangent functions are only positive. Sine , Cosine , cosecant and secant all functions are negative. Now we can write:   $\begin{array}{l}\text{sin}\text{θ}=-\text{y};\csc \text{θ}=\frac{1}{\sin \text{θ}}=-\frac{1}{\text{y}}\\ \cos \text{θ}=-\text{x};\sec \text{θ}=\frac{1}{\cos \text{θ}}=-\frac{1}{\text{x}}\\ \tan \text{θ}=\frac{\text{sin}\text{θ}}{\text{cos}\text{θ}}=\frac{\text{y}}{\text{x}};\cot \text{θ}=\frac{1}{\tan \text{θ}}=\frac{x}{\text{y}}\end{array}$

Calculation:

  $\text{sin}\text{θ}=-\text{y};\csc \text{θ}=-\frac{1}{\text{y}};\cos \text{θ}=-\text{x};\sec \text{θ}=-\frac{1}{\text{x}};\tan \text{θ}=\frac{\text{y}}{\text{x}};\cot \text{θ}=\frac{x}{\text{y}}$

Thus, tan $\text{θ}$ and cot $\text{θ}$ are positive only. Sine, cosine, cosecant and secant all are negative.