Lesson 6 Hidden Identities Practice Understanding

Jump Start

Examine each of the following equations. Share with a partner a reason why each one doesn’t belong in this set of equations.

$(x - 3) (x + 2) = 0$

$\frac{4}{3} x = 2 x$

$x^{2} - 5 x + 6 = 0$

$\sin (x) = 1$

Learning Focus

Solve trigonometric equations using identities and graphs.

What mathematical tools do I use to solve trigonometric equations?

Open Up the Math: Launch, Explore, Discuss

Note: Because trigonometric functions are periodic, trigonometric equations often have multiple solutions. Typically, we are only interested in the solutions that lie within a restricted interval, usually the interval from $0$ to $2 π$ . In this task, you should find all solutions to the trigonometric equations that occur on $[0, 2 π]$ .

To sharpen their trigonometry skills, Alyce, Javier, and Veronica are trying to learn how to solve some trigonometric equations in a math refresher text that they found in an old trunk one of the adults had brought to the archeological site. Here is how each of them thought about one of the problems:

Solve: $\cos (\frac{π}{2} - θ) = \frac{1}{2}$

Alyce: I used the inverse cosine function.

Javier: I first used an identity, and then an inverse trigonometric function. But it was not the same inverse trigonometric function that Alyce used.

Veronica: I graphed $y_{1} = \cos (\frac{π}{2} - θ)$ and $y_{2} = \frac{1}{2}$ on my calculator. I seem to have found more solutions.

1.

Using their statements as clues, go back and solve the equation the way that each of the friends did.

2.

How does Veronica’s solutions match with Alyce and Javier’s? What might be different?

Solve each of the following trigonometric equations by adapting Alyce and Javier’s strategies: that is, you may want to see if the equation can be revised using one of the trigonometric identities you learned in the previous task; and once you have isolated a trigonometric function on one side of the equation, you can undo that trigonometric function by taking the inverse trigonometric function on both sides of the equation. Once you have a solution, you may want to check to see if you have found all possible solutions on the interval $[0, 2 π]$ by using a graph as shown in Veronica’s strategy.

3.

$\sin (- θ) = - \frac{1}{2}$

4.

$\cos θ \cdot \tan θ = \frac{\sqrt{3}}{2}$

5.

$\sin (2 θ) = \frac{\sqrt{2}}{2}$

Ready for More?

Solve the following equation for $θ$ on the interval $[0, 2 π]$ :

$\cos (3 θ) \cdot \tan (3 θ) = \frac{\sqrt{3}}{2}$

Takeaways

Sometimes it is useful to find the solutions to a trigonometric function within a restricted domain, such as $[0, 2 π]$ , since

Typically, there are two solutions to a trigonometric equation on the interval from $0$ to $2 π$ , with the following exceptions:

We have extended the method for solving trigonometric equations to include:

Vocabulary

restricted domain
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we extended the process for solving trigonometric equations to include looking for trigonometric identities that might change the trigonometric expressions to simpler forms. We also identified ways to determine how many solutions to the trigonometric equation occur in the interval from 0 to $2 π$ .

Retrieval

1.

Use the diagram to help you find the two angles, $0 < θ \leq 2 π$ , that are solutions to the equation. Round your answers to $2$ decimals. (Your calculator should be set in radians.)

$\sin θ = \frac{2}{3}$

2.

Find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s$ . $(s = r θ)$

Round your answer to $4$ decimal places.

Radius = $13 in.$ Arc Length = $11 in.$

Angle measure in radians: