# Determine the values of the trigonometric functions of angle a if P is a point on terminal side of a .The coordinates of P are (3,4); (-3,4); (-1,-3) .

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### 2 Answers

The point P has the coordinates (3,4), (-3,4) and (-1, -3)

To determine the angle the line joining the origin with P makes with the x axis..

Let us draw the perpendicular from P to the x axis.

Then (OA , AP) are the x and y coordinates of P.

P(3,4)):

tan XOP = tan AOP = AP/OA= 4/3.

Angle XOP = arctan (4/3) = 53.13 deg.

P(-3,4):

Angle XOP = arc tan (4/-3) = 126.87 deg.

P((-1, -3)

Angle XOP = arctan (-3/-1) = 251.57 deg

We know that the definitions of the trigonometric functions are:

sin a = y/r

cos a = x/r

tan a = y/x

cot a = x/y

sec a = r/x

csc a = r/y

We'll calculate r using the Pythagorean theorem in a right angle triangle:

r^2 = x^2 + y^2

r = sqrt (x^2 + y^2)

We'll calculate the values for the trigonometric functions and we'll choose the smallest positive angle in the standard position for P(3,4).

r = sqrt (3^2 + 4^2)

r = sqrt(9+16)

r = sqrt 25

r = 5

sin a = 4/5 ; cos a = 3/5

tan a = sin a/cos a

tan a = (4/5)/(3/5)

tan a = 4/3

cot a = 1/tan a

cot a = 3/4

sec a = 5/3

csc a = 5/4

We'll calculate the values for the trigonometric functions and we'll choose the smallest positive angle in the standard position for P(-3,4).

r = sqrt [(-3)^2 + 4^2]

r = sqrt(9+16)

r = sqrt 25

r = 5

sin a = 4/5 ; cos a = -3/5

tan a = sin a/cos a

tan a = (4/5)/(-3/5)

tan a = -4/3

cot a = 1/tan a

cot a = -3/4

sec a = -5/3

csc a = 5/4

We'll calculate the values for the trigonometric functions and we'll choose the smallest positive angle in the standard position for P(-1,-3).

r = sqrt [(-1)^2 + (-3)^2]

r = sqrt(1+9)

r = sqrt 10

sin a = -3/sqrt10

sin a = -3*sqrt10/10

cos a = -1/sqrt10

cos a = -sqrt10/10

tan a = (-3*sqrt10/10)/( -sqrt10/10)

tan a = 3

cot a = 1/3

sec a = -sqrt10

csc a = -sqrt10/3