Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Trigonometric Ratios and Trigonometric Functions

Let  ABC  be a right angled triangle, with   M < ACB=\dfrac{\pi}{2}, let  M< ABC=\theta

Then,  0 < \theta< \dfrac{\pi}{2} .

The lengths AC, BC, \text{ and }AB are known as  P \text {( perpendicular)}, b \text {(base)}, \text {and } h \text{ (hypothenuse)} respectively.

Definiton 1

(i)  \sin\theta=\dfrac{P}{h}=\dfrac{AC}{AB}

 (ii)  \cos\theta=\dfrac{b}{h}=\dfrac{BC}{AB}

(iii)  \tan\theta=\dfrac{P}{b}=\dfrac{AC}{BC}

(iv)  \csc\theta=\dfrac{h}{p}=\dfrac{AB}{AC}

(v)  \sec\theta=\dfrac{h}{b}=\dfrac{AB}{BC}

(vi)  \cot\theta=\dfrac{b}{P}=\dfrac{BC}{AC}

We also defined:

(i)  \tan\theta=\dfrac{\sin\theta}{\cos\theta}

(ii)  \cot\theta=\dfrac{\cos\theta}{\sin\theta}\text { or }\dfrac{1}{\tan\theta}

(iii)  \sec\theta=\dfrac{1}{\cos\theta}

(iv)  \csc \theta=\dfrac{1}{\sin\theta}

Definition 2

For angle-measure  0^o  we defined:

 \sin 0^o= 0 ,

 \cos 0^o= 1 ,

 \tan 0^o= 0

 \sec 0^o= \dfrac{1}{\cos O^o}=1

 \to \cot 0^o  \text { and } \csc 0^o  are not defined.

Definition 3

For angle- measure  \dfrac{\pi}{2} we defined  \sin \dfrac{\pi}{2}=1,\quad\cos \dfrac{\pi}{2}=0 \quad \csc  \dfrac{\pi}{2}=\dfrac{1}{\sin \dfrac{\pi}{2}} =1, \quad \cot\dfrac{\pi}{2}=\dfrac{\cos\dfrac{\pi}{2}}{\sin\dfrac{\pi}{2}}=0

\to \tan\dfrac{\pi}{2} \text { and } \sec \dfrac{\pi}{2} are not defined.

Note:

 \to \sin\dfrac{\pi}{2},  \cos \dfrac{\pi}{2}, \cot\dfrac{\pi}{2,},\text{ and }\csc\dfrac{\pi}{2}  have not defined as rations of lengths. So we do not use the term ‘trigonometric ratio’ for them. We shall replace the term by more general term ‘trigonometric function.’

Trigonometric Function for  \theta\in R

Let  (r, \theta)_{x}\text{ and }(r,\theta)_{y} denote respectively the x-coordinates and y-coordinates of a point whose polar coordinates are  (r, \theta); r > 0, \text{ and } \theta\in R

Definitions

(i) Sine : R\longrightarrow [-1,1]; \quad\sin\theta=\dfrac{(r,\theta)_{y}}{r}; \quad r> 0,\theta\in R

(ii) Cosine:  R\longrightarrow [-1,1];\quad\cos\theta=\dfrac{(r,\theta)_{x}}{r}; r > 0, \theta\in R

(iii) Tangent:  R-\left\{ (2n+1)\dfrac{\pi}{2}:n\in Z \right\}\longrightarrow R;

 \tan\theta=\dfrac{(r,\theta)_{y}}{(r,\theta)_{x}} \text { or } \dfrac{\sin\theta}{\cos\theta};r > 0

 \theta\in R-\left\{ (2n+1)\dfrac{\pi}{2} :n\in Z\right\}

(iv) Cotangent:  R-\left\{ n\pi:n\in Z \right\}\to R;

 \cot\theta=\dfrac{(r,\theta)_{x}}{(r,\theta)_{y}} \text { or } \dfrac{\cos\theta}{\sin\theta}; \quad r > 0,\theta \in R-\left\{ n\pi:n\in Z \right\}

(v) Secant:  R-\left\{ (2n+1)\dfrac{\pi}{2}:n\in Z \right\}\longrightarrow R-(-1,1);

 \sec\theta=\dfrac{r}{(r,\theta)_{x}}\text { or } \dfrac{1}{\cos\theta};\quad r > 0, \theta\in R-\left\{ (2n+1)\dfrac{\pi}{2}:n\in Z \right\}

(vi) Cosecant: R-\left\{ n\pi:n\in Z \right\}\longrightarrow R-(-1,1)

 \csc\theta=\dfrac{r}{(r,\theta)_{y}}\text { or } \dfrac{1}{\sin\theta}; \quad r > 0, \theta \in R-\left\{ n\pi:n\in Z \right\}

Note:

By taking  r=1 in the above definitions we get the trigonometric functions from a unit circle.

Domain and Range of Trigonometric Functions

  Domain Range
(i) \sin \theta  R  [-1,1]
(ii) \cos \theta  R [-1,1]
(iii) \tan \theta R-\left\{ (2n+1)\dfrac{\pi}{2}: n \in Z \right\}  R
(iv)  \cot \theta  R -\{n \pi : n \in Z\} R
(v)  \sec \theta  R-\left\{ (2n+1)\dfrac{\pi}{2}: n \in Z \right\}  R-(-1,1)
(vi) \csc \theta  R-\{n \pi : n \in Z\}  R-(-1,1)
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