Lesson 9 More Hidden Identities Practice Understanding

Learning Focus

Solve trigonometric equations strategically.

What additional tools and ideas can I use to solve trigonometric equations?

Open Up the Math: Launch, Explore, Discuss

Note: Because trigonometric functions are periodic, trigonometric equations generally have multiple solutions. In this task, you should find all solutions to the trigonometric equations by representing a sequence of solutions using the form $answer \pm k π n$ , where $k π$ represents the interval between successive solutions.

Previously, Javier observed that he might need to change the form of the trigonometric expressions in a trigonometric equation before he could use an inverse trigonometric function to solve the equation. Changing the form of a trigonometric expression involves looking for trigonometric identities that apply to the expression. Sometimes you have to manipulate the expression algebraically before you can identify a trigonometric identity that might apply.

Here is a summary of how Javier would have solved a problem previously. You can use his strategy to solve the remaining problems in this task.

1.

Solve: $2 \sin (x) \cos (x) + \frac{\sqrt{3}}{2} = \sqrt{3}$

a.

Idea #1: Do you see a trigonometric identity? If so, write the equation in an equivalent form using the identity. If not, can you manipulate the trigonometric equation algebraically?

Which approach do you think Javier would need to take for his first step to solve this problem: use an identity or manipulate the equation algebraically?

b.

Apply your thought:

c.

Idea #2: Repeat idea #1 until you have isolated the trigonometric function on one side of the equation.

Apply idea #2:

d.

Idea #3: Apply the inverse trigonometric function to both sides of the equation, which will give an equation of the form: an expression = an angle; solve this equation for the variable.

Apply idea #3:

e.

Idea #4: This process finds one of an infinite number of solutions. Since trigonometric functions are periodic, adding or subtracting multiple increments of the period will generate new solutions. Write this infinite set of solutions in the form $answer + q π k$ , where $q π$ represents the interval between successive solutions.

Apply idea #4:

f.

Idea #5: Since the inverse trigonometric function used in #3 is a function, we only get one unique value as a solution to the inverse trigonometric expression, while there may be another solution in a different quadrant on the unit circle. Find this alternate solution, and the corresponding infinite set of solutions to the equation.

Apply idea #5:

g.

Hints for solving trigonometric equations: Graphs of trigonometric functions and the unit circle can be used as tools to help reason about the solutions to a trigonometric equation.

Illustrate how a graph and the unit circle can help you find the solutions to this equation:

Graph:

Unit Circle:

Solve the following trigonometric equations by applying the ideas and hints described above.

2.

$2 \cos (3 x) = \sqrt{2}$

3.

$\sin^{2} (x) - \sin (x) - 2 = 0$

Pause and Reflect

4.

$2 \cos^{2} (x) - \sqrt{3} \cos (x) = 0$

5.

$2 \sin (2 x) - 3 \sin (x) = 0$

6.

$(\sin {(x) + \cos (x))}^{2} = \frac{3}{2}$

Ready for More?

Solve: $\sin (2 x) - \tan (x) = 0$

Takeaways

New issues to attend to when solving trigonometric equations:

Lesson Summary

In this lesson, we extended our strategies for solving trigonometric equations to include equations with different multiples of the variable inside trigonometric expressions, equations that could be solved exactly using the unit circle diagram, and those that required using the inverse trigonometric features of the calculator, and trigonometric equations that behaved like quadratic equations, requiring expressions to be factored.

Retrieval

1.

A complex number has been graphed as a vector.

Rewrite the complex number as a point in the form $(a, b)$ .

2.

Graph the vector $(4, i)$ on the complex plane.

3.

Find $| - 5 + 3 i |$ and $| 4 + i |$ .

4.

Use the graph of $y = 3 (\cos 3 x) - 2$ to find all of the values for $x$ when $y = - 2$ . Write your answer(s) in the form $\pm 2 π k$ , where $2 π k$ represents the interval between successive solutions.

5.

Often when we graph in radians, the steps on the $x$ -axis have numbers instead of multiples of $π$ . Replace $- π$ , $- \frac{π}{2}$ , $\frac{π}{2}$ , $π$ on the $x$ -axis with the numbers they represent.