Pearson Algebra 2 Common Core, 2011
PA
Pearson Algebra 2 Common Core, 2011 View details
2. Solving Trigonometric Equations Using Inverses
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Exercise 61 Page 917

Practice makes perfect
a We want to write three trigonometric equations such that the complete solution of each is We will first find one trigonometric equation that has as its only solution in the interval To do so, let's consider a unit circle with angle equal to
Unit Circle
Recall that the length of the radius of a unit circle is Also, the coordinate of the point is and the coordinate of is
Based on the unit circle, we can determine the values of and where is equal to
We will now examine the unit circle again to see if the equations and have any solutions other than

Solution(s) of

Let's consider Remember, the coordinate of any point represents the sine value, so we will look for the point(s) whose coordinate is

Unit Circle
It can be seen that the equation has one solution other than in the interval
Keep in mind that if we add or subtract a multiple of radians, the terminal side of the angle will be in the same position. Therefore, the sine of the resulting angles will also be solutions to the equation.
Notice that the complete solution of does not only include Therefore, is not an example of the type of trigonometric equation that we want to find.

Solution(s) of

Let's now consider Since the coordinate of any point on the unit circle represents the cosine value, we will look for the point(s) whose coordinate is

Unit Circle

Note that is the only point whose coordinate is This implies that is the only solution to the equation in the interval Therefore, we can write the complete solution of the equation as where is an integer. This is exactly what we are looking for!

Conclusion

We found that one of the trigonometric equations with the complete solution is
We can write a second example by multiplying both sides of the Equation I by
Let's now write a third example by multiplying both sides of the Equation I by
Note that answers may vary. It is possible to write many different trigonometric equations with the complete solution of by multiplying both sides of the Equation I by any non-zero number.
b There were two main steps that we took to find the equations in Part A.
  1. Using a unit circle, we determined the first example of an equation, with the complete solution
  2. We obtained other equations by multiplying both sides of by any non-zero number.