Lesson 7E Double Identity Solidify Understanding
Find an expression that will give the length of the other two sides of the right triangle using the given constants
Derive trigonometric identities for the sum or difference of two angles.
Why might some people think that
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
Do you think this is a true statement?
Why or why not?
Examine the diagram. Figure
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
Once you have an identity for
You can find an identity for
Now you can also complete this identity using reasoning similar to what you did in problem 3.
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.
Ready for More?
Derive alternative forms of the double angle identity for
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:
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.
Find the two angles,
Find the length of an arc given that