a We want to write three trigonometric equations such that the complete solution of each is π+2πn. We will first find one trigonometric equation that has π as its only solution in the 0≤θ<2π. To do so, let's consider a with θ equal to π.
Recall that the length of the of a unit circle is
1. Also, the
x- of the
P is
cosθ and the
y-coordinate of
P is
sinθ.
P(x,y)=P(cosθ,sinθ)
Based on the unit circle, we can determine the values of
cosθ and
sinθ where
θ is equal to
π.
P(-1,0)=P(cosθ,sinθ)⇓cosθ=-1sinθ=0
We will now examine the unit circle again to see if the equations
cosθ=-1 and
sinθ=0 have any solutions other than
π.
Solution(s) of sinθ=0
Let's consider sinθ=0. Remember, the y-coordinate of any point represents the value, so we will look for the point(s) whose y-coordinate is 0.
It can be seen that the equation
sinθ=0 has one solution other than
π in the interval
0≤θ<2π.
sinθ=0⇒θ1=0 or θ2=π
Keep in mind that if we add or subtract a multiple of
2π radians, the will be in the same position. Therefore, the sine of the resulting angles will also be solutions to the equation.
0+2πn radiansandπ+2πn radianswhere n is any integer
Notice that the complete solution of
sinθ=0 does
not only include
π+2πn. Therefore,
sinθ=0 is
not an example of the type of trigonometric equation that we want to find.
Solution(s) of cosθ=-1
Let's now consider cosθ=-1. Since the x-coordinate of any point on the unit circle represents the value, we will look for the point(s) whose x-coordinate is -1.
Note that P(-1,0) is the only point whose x-coordinate is -1. This implies that π is the only solution to the equation cosθ=-1 in the interval 0≤θ<2π. Therefore, we can write the complete solution of the equation cosθ=-1 as π+2πn, where n is an . This is exactly what we are looking for!
Conclusion
We found that one of the trigonometric equations with the complete solution
π+2πn is
cosθ=-1.
Equation I: cosθ=-1
We can write a second example by multiplying both sides of the Equation I by
2.
Equation II: 2cosθ=-2
Let's now write a third example by multiplying both sides of the Equation I by
3.
Equation III: 3cosθ=-3
Note that answers may vary. It is possible to write many different trigonometric equations with the complete solution of
π+2πn by multiplying both sides of the Equation I by any non-zero .