Trigonometry
Converting Rectangular & Polar Linear Equations
Converting a Polar Linear Equation to Rectangular Form
The polar form of a linear equation is p
=
rcos
θ
−
φ
(
), where p and
φ
are constants
dictating the angle and length of the normal to the line.
To convert a Polar Linear Equation to Rectangular, we need to use the COSINE ANGLE
SUM/DIFFERENCE identity:
cos A
±
B
(
)
=
cos Acos B∓sinAsin B
For example, convert 1=rcos
θ
−
30°
(
) to standard rectangular form.
Apply the identity:
1=rcos
θ
−30°()
1=rcos
θ
cos30°+sin
θ
sin30°
()
Evaluate cos30° and sin30°:
1=rcos
θ
⋅3
2+sin
θ
⋅1
2
⎛
⎝
⎜
⎞
⎠
⎟
Distribute r:
1=rcos
θ
⋅3
2+rsin
θ
⋅1
2
Remember the definitions of sine and cosine.
sin
θ
=y
r, so rsin
θ
=
y
cos
θ
=x
r, so rcos
θ
=
x
In the equation, replace rcos
θ
with x and rsin
θ
with y:
1=x⋅3
2+y⋅1
2
To put this rectangular equation in standard form, get rid of the fractions:
2=x3+y
x3+y−2=0
