Now read the following article carefully to clear all of your doubts (please make it spread/ available to all your real friends for success) 

We can relate sine and cosine to coordinates on a graph. The ordinate, or y-value, on a graph is the sine value for a given angle. The quadrant that the angle is in, therefore, is very important in this matter. Y-values are positive in the first and second quadrants; therefore, sine values will be positive in first and second quadrants. Y-values are negative in third and fourth quadrants; therefore, sine values will be negative i third and fourth quadrants.

 Abscissa, or x-values, would be our cosine values. The value of the abscissa will determine the value of the cosine of the given angle. Since x-values are positive in the first and fourth quadrants, cosine values will be positive in the first and fourth quadrants. Since x-values are negative in the second and third quadrants, cosine values will be negative in the second and third quadrants.

 Trigonometry refers only to right triangles. All triangles have a total of 180 degrees. If a triangle is located in the second, third, or fourth quadrant, it needed to rotate there from the first quadrant. Since rotation forms a circle, and since circles have 360 degrees, we need to determine the equivalent angle of rotation for a given triangle. For example, if a tire rotates 450 degrees, the position that it would be in has an equivalent position between 0 and 360 degrees. To determine that equivalent (coterminal)angle, we subtract 360 degrees to get the answer 90 degrees. If the tire reverses itself, it would have negative angles. To determine the coterminal angle, we add 360 degrees to the negative angle. For example, if we reverse the tire -632 degrees, we add 360 degrees to see it reversed -272 degrees. We add 360 degrees again to get an angle of 88 degrees.  To determine which quadrant certain angles lie in after rotations have been made, refer to the following chart:

Between 0 & 90 degrees = Quadrant I

Between 90 & 180 degrees = Quadrant II

Between 180 & 270 degrees = Quadrant III

Between 270 & 360 degrees = Quadrant IV

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 We can identify sine and cosine by the abscissa (x-value) and ordinate (y-

value) for a given co-ordinate. Let's picture a graph with cor-ordinates 

(x,y). The opposite leg is the y value,; therefore, the sine of a given 

angle theta is equal to the ordinate (y-value). The adjacent leg is the x-

value; therefore, the cosine of a given angle theta is equal to the abscissa 

(x-value). In the first quadrant, all values are positive; therefore, sine 

and cosine values will always be positive. However, as we move into the 

second quadrant, cosine becomes negative since x-values are negative in that 

quadrant. The sine values remain positive, though. Moving into the third 

quadrant, both the x-values and y-values are negative; therefore, all sine 

and cosine values are negative here. In the fourth quadrant, x-values again 

become positive so cosine values become positive again. Y-values remain 

negative here, so sine values remain negative here. Therefore, if we have a 

rotation greater than 90 degrees, we can determine the coterminal angle and 

evaluate the sine and cosine values for the coterminal angles. They will be 

the same for the angles greater than 90 degrees, except we need to keep in 

mind which quadrants have positive and negative sine/cosine values. See 

above for the angles that lie in a given quadrant.

 Tangent can be defined as opposite over adjacent. Another way of writing 

tangent is to use trigonometric functions. The function that refers to the 

opposite leg is sine while the function that refers to the adjacen tleg is 

cosine. Therefore, tangent is sine/cosine. Since sine and cosine are 

positive in the first quadrant, tangent is positive in the first quadrant. 

Since sine is positive an dcosine is negative in the second quadrant, 

tangent is negative in the second quadrant. Since both sine and cosine are 

negative in the third quadrant, tangent will be positive in the third 

quadrant. Since sine is negative in the fourth quadrant and cosine is 

positive in the fourth quadrant, tangent will be negative in the fourth 

quadrant. Just remember, All Students Take Class and you will remember the 

positive functions for the various quadrants.

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 If we were to draw a circle on a graph with a radius of 1, we would have 

coordinates of (1,0), (0,1), (-1,0), and (0,-1). These coordiantes 

correspond with the angles of 0 degrees/360 degrees = (1,0); 90 degrees = 

(0,1); 180 degrees = (-1,0); and 270 degrees = (0,-1). Using the method above

we can identify the sine, cosine, and tangent values for each of these 

angles.

ANGLE           SINE (y-value)     COSINE (x-value)    TANGENT (sine/cosine)

0 degrees             0                 1                0/1 = 0

90 degrees            1                 0                1/0 = undefined

180 degrees           0                -1                0/-1 = 0

270 degrees          -1                 0               -1/0 = undefined

360 degrees           0                 1                0/1 = 0

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 If we were to draw an equilateral triangle on a graph and bisect it, we 

would create two right triangles with the following angle measures - 30 

degrees, 60 degrees, and 90 degrees. This is a special case. If we use the 

Pythagorean formula and plug in the distance of 1 (hypotenuse) and 0.5 (one 

leg opposite 30 degrees), we would have a value of radical 3/2 for the leg 

adjacent to 30 degrees. Using the values of 1, 0.5, and radical 3/2, we can 

determine the sine, cosine, and tangent values for 30 and 60 degrees.

ANGLE        SINE (opposite leg)  COSINE (adjacent leg) TANGENT (sine/cosine)

30 degrees         0.5                 radical 3/2      0.5/(rad 3/2)= rad3/3

60 degrees      rad 3/2                  0.5            (rad 3/2)/0.5= rad 3

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 If we were to draw an isosceles right triangle, we would have angle 

measures of 45 degrees, 45 degrees, and 90 degrees. This is a special case. 

If our hypotenuse is 1, we can use the Pythagorean formula to determine the 

values of the other two legs. They would each measure radical 2/2. The sine, 

cosine, and tangent values for 45 degrees can then be determined.

ANGLE      SINE (opposite leg)   COSINE (adjacent leg)  TANGENT (sine/cosine)

45 degrees    radical 2/2            radical 2/2      (rad 2/2)/(rad 2/2) = 1

 THESE VALUES MUST BE MEMORIZED!!!

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 To find the values for any given angle, we simply punch into the calculator 

the function value that we are seeking and the angle we want. Then push 

enter. This is true even if we are dealing with angles greater than 90 

degrees.

 However, we still need to know the reference angle in the first quadrant 

for any angle in another quadrant. To do this, follow these rules:

QUADRANT         ANGLE RANGE          FORMULA

II               90-180 degrees       180 degrees - theta

III              180-270 degrees      theta - 180 degrees

IV               270-360 degrees      360 degrees - theta

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 We can measure angles in other units besides degrees. Another unit we use 

is radians. One radian is equal to the measure of the radius of a circle 

when there is a central angle drawn in the circle. The angle we are 

referring to is the arc intercepted by the central angle. The measure of the 

angle in radians can be determined by dividing the length of the intercepted 

arc by the length of the radius. 

                   length of the intercepted arc

Angle measure = --------------------------------------

                    length of the radius

 From the above formula, we can determine a relationship between a radian 

and a degree. Let's look at a semicircle. A semicircle has an arc equal to 

half the circumference of the circle. Therefore, the arc equals (pi)(radius).

The angle of the semicircle is 180 degrees. To find the number of radians, 

we can use the formula above:

                   (pi)(radius)                     pi

180 degrees = ------------------- => 180 degrees = -----, or 180 degrees = pi

                length of radius                     1   

 How many degrees is one radian equal to? Divide 180 degrees by pi and we 

get 57.3 degrees.

 To change from degrees to radians, we use the proportion:

measure of angle in degrees         180 degrees

----------------------------   =   -------------

measure of angle in radians              pi

For example, 20 degrees is equal to:

20 degrees          180 degrees        20pi = 180 radians => 20pi

----------   =   ---------------- =>                         ---- = radians

radians                pi                                     180

pi

-- = radians

9 

 To change from radians to degrees, we simply substitute 180 degrees for pi 

and do the math.

pi                180

-- radians =>    ----- = 15 degrees

12                12

The following table should be memorized:

DEGREE           RADIAN

0                0

30              pi/6

45              pi/4

60              pi/3

90              pi/2

180             pi

270             3pi/2

360             2pi

 To determine the trigonometric function values for sine, cosine, and 

tangent for radian measure, we switch our calculators from degrees to 

radians and punch in the numbers as we did before. When we see pi, we use 

radian. Otherwise, we use degrees.

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 Each trigonometric function has a reciprocal function. The reciprocal 

function of sine is cosecant (1/sine theta). The reciprocal function of 

cosine is secant (1/cosine theta). The reciprocal function of tangent is 

cotangent (1/tangent theta). Our calculators do not have these function keys 

on them so we need to punch in 1/ whatever the function is we need here for 

the correct reciprocal. This is true regardless of whether or not we need 

degrees or radians.

 In what quadrants are these reciprocals positive? In the same quadrants as 

the function sthey are derived from.

QUADRANT          FUNCTION          VALUE

I                  all               positive

II