Lesson 7 Double Identity Solidify Understanding

Jump Start

Find an expression that will give the length of the other two sides of the right triangle using the given constants $N$ and $β$ .

Learning Focus

Derive trigonometric identities for the sum or difference of two angles.

Why might some people think that $\sin (α + β) = \sin α + \sin β$ , and what evidence might I provide that this statement is incorrect? Is there an expression that is equivalent to $\sin (α + β)$ ?

Open Up the Math: Launch, Explore, Discuss

Sum and Difference Identities

Sometimes it is useful to be able to find the sine and cosine of an angle that is the sum of two consecutive angles of rotation. In the upcoming diagram, point $P$ has been rotated $α$ radians counterclockwise around the unit circle to point $Q$ , and then point $Q$ has been rotated an additional $β$ radians counterclockwise to point $R$ . In this task, you will examine how the sine and cosine of angle $α$ , angle $β$ , and the sum of the two angles, angle $α + β$ , are related.

1.

Do you think this is a true statement? $\sin (α + β) = \sin α + \sin β$

Why or why not?

Examine the diagram. Figure $O A C D$ is a rectangle.

Can you use this diagram to state a true relationship that completes this identity? (Your teacher has some hint cards if you need them, but the basic idea is to label all of the segments on the sides of rectangle $O A C D$ using right triangle trigonometric relationships.) Based on congruent line segments labeled with trigonometric measures in the diagram, complete the following identity:

2.

$\sin (α + β) =$

Once you have an identity for $\sin (α + β)$ you can find an identity for $\sin (α - β)$ algebraically. Begin by noting that $\sin (α - β) = \sin (α + (- β))$ and apply the identity you found in problem 2, along with the identities you explored previously: $\sin (- θ) = - \sin (θ)$ and $\cos (- θ) = \cos (θ)$ .

3.

$\sin (α - β) =$

You can find an identity for $\cos (α + β)$ in the diagram also. Since $\overset{―}{O A} ≅ \overset{―}{D C}$ , and $D C = D R + R C$ , using trigonometry to determine the lengths of segments $O A$ , $D R$ , and $R C$ will reveal this relationship. (Again, your teacher has hint cards if you need them.)

4.

$\cos (α + β) =$

Now you can also complete this identity using reasoning similar to what you did in problem 3.

5.

$\cos (α - β) =$

Pause and Reflect

The following identities are known as the double angle identities, but they are just special cases of the sum identities you found in the previous problems.

6.

$\sin (2 α) = \sin (α + α) =$

7.

$\cos (2 α) = \cos (α + α) =$

Ready for More?

Derive alternative forms of the double angle identity for $\cos (2 α)$ by using the Pythagorean identity $\sin^{2} (α) + \cos^{2} (α) = 1$ .

Takeaways

Trigonometric expressions can be manipulated by applying

Today, we added the following to our collection of trigonometric identities:

The sum and difference identities:

The double angle identities:

Alternative forms:

Vocabulary

double angle identities
sum and difference identities
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we expanded our list of trigonometric identities to include identities for finding the sine or cosine of angles that are the result of adding two angles together or subtracting one angle from another one. If the angles are the same size, then we can use these sum identities to find the sine and cosine of an angle that is twice as big as a given angle.

Retrieval

1.

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.)

$\cos θ = - \frac{9}{41}$

2.

Find the length of an arc given that $r = 68 in.$ and $θ = \frac{3 π}{4}$ . (Write your answers with $π$ in it. Then use your calculator to find the approximate length of the arc to $2$ decimal places.)