Fundamental Trigonometric Ratios

1) Sin\theta =  \frac{{perpendicular}}{{hypotenuse}} = \frac{y}{r}

2) Cos\theta = \frac{{base}}{{hypotenuse}} = \frac{x}{r}   

3) Tan\theta = \frac{{perpendicular}}{{base}} =  \frac{y}{x} = \frac{{Sin\theta }}{{Cos\theta }}

4) Cot\theta = \frac{{base}}{{perpendicular}} =  \frac{x}{y} = \frac{{Cos\theta }}{{Sin\theta }}

5) Sec\theta = \frac{{hypotenuse}}{{base}} = \frac{r}{x} =  \frac{1}{{Cos\theta }}

6) Co\sec \theta = \frac{{hypotenuse}}{{perpendicular}} =  \frac{r}{y} = \frac{1}{{Sin\theta }}

7) Signs of trigonometric Ratios
    I QUADRANT: A, means all trigonometric ratios are positive.
   II QUADRANT: S, means Sin and Co\sec are positive all others are negative.
  III QUADRANT: T, means Tan and Cot are positive all others are negative.
  IV QUADRANT: C, means Cos and Sec are positive all others are negative.

NOTE: (1) Clockwise; we read ACTS
            (2) Anticlockwise; we read ASTC (All Silver Tea Cups)

\theta in Quadrant

Sin\theta

Cos\theta

Tan\theta

Cot\theta

Sec\theta

Co\sec \theta

I

 +

 +

 +

 +

 +

 +

II

 +

 -

 -

 -

 -

 +

III

 -

 -

 +

 +

 -

 -

IV

 -

 +

 -

 -

 +

 -

8) Trigonometric Ratios of Special Angles:

\theta

Sin\theta

Cos\theta

Tan\theta

Cot\theta

Sec\theta

Co\sec \theta

{0^ \circ }

0

1

0

\infty

1

\infty

{30^ \circ }

\frac{1}{2}

\frac{{\sqrt 3 }}{2}

\frac{1}{{\sqrt 3 }}

\sqrt 3

\frac{2}{{\sqrt 3 }}

2

{45^ \circ }

\frac{1}{{\sqrt 2 }}

\frac{1}{{\sqrt 2 }}

1

1

\sqrt 2

\sqrt 2

{60^ \circ }

\frac{{\sqrt 3 }}{2}

\frac{1}{2}

\sqrt 3

\frac{1}{{\sqrt 3 }}

2

\frac{2}{{\sqrt 3 }}

{90^ \circ }

1

0

\infty

0

\infty

1

 
RULE: Write 0,1,2,3,4 divide by 4 then taking square roots and write the resulting number in column of Sin\theta  .