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Verifying Trigonometric Identities can either be a thought provoking exploration of trigonometric identities or a frustrating exercise of seemingly useless facts… it’s all about how you sell it. If students have a solid foundation in the basic trigonometric identities and a fundamental understanding of the relationship between the unit circle and the special right triangles then this unit can be enjoyable for both the teacher and the students. Since, proofs have been largely removed from the current geometry and or tenth grade math curriculum some groundwork may be needed to help train students on how to start a logical approach to solving, simplifying, and verifying identities.

Here’s a nice little intro to verifying trig identities.

http://goanimate.com/videos/0XXFlEW-cu8Y

The easiest identities to introduce after the basic identities (which should have already been covered in the intro to trig unit) are probably the Pythagorean Identities. These identities can easily be connected back to the unit circle and the coordinate values of cosine (x) and sine (y) and the radius of one. After establishing that sin²(x) = cos²(x) = 1, it is pretty easy to get students to derive the other forms of the Pythagorean Identities. The attached task will help students work through this knowledge. You can allow students to work through the task first and discover the identities for themselves and then use the flipchart to reinforce what they have learned or vice versa. If you are pressed for time I would suggest using the flipchart first and then introducing the task. You should be able to complete this portion of the unit in one to two 90-minute block classes.

Exploring the Pythagorean Identities (STUDENT, TEACHER)

Pythagorean Identities flipchart to pair with instruction.

Next I would introduce the sum and difference identities. Students usually grasp this identity quickly especially since it allows them to use numeric values and gives them a little break from those pesky variables.

I would have them work individually for at least 20 minutes prior to allowing them to work in pairs or groups. This portion of the unit should only take about one 90-minute class period.

Sum and Difference Identities (STUDENT, TEACHER)

Sum and Difference Identities flipchart to pair with instruction.

Now students are equipped with the knowledge needed to begin simplifying, solving and proving identities. “Where’s the Identity” allows students to analyze identities graphically, numerically and algebraically students learn multiple ways to verify their answers. Graphing calculators or an online graphing calculator application will be needed to do this activity. I would also encourage students to download a graphing calculator app if they have not already done so and/or don’t own a graphing calculator.

This task can be given as a traditional task or you can set up stations (two of graphing, algebraic and numeric, as well as a computer station) or give it to students as a traditional task. If given as a traditional task you can have different groups of students work on the different components and present their findings to the class. There are a lot of ways to differentiate this particular activity, so be creative!

Where’s the Identity (STUDENT, TEACHER)

“Establishing the Identities” is a culminating activity, which uses all of the information, gained through out this unit. You can give it as an individual assignment, allow students to work in pairs or small groups or make it a more interactive activity by using the white boards.

I typically give my students their own whiteboards and markers during this portion so I can see their thought process and it is easier to make corrections to missteps in the problem…not to mention the fact that my students LOVE writing on the whiteboards. I always give a quick spill about the boards being used for instructional purposes only and failure to comply resulting in the board and marker being confiscated and the offender receiving a daily participation grade of a zero. Making this announcement at the beginning of class generally curtails any major classroom disturbances with the boards. Me and the kids usually have a blast doing this task with the white boards because it's a fun way to see how different people approach verifying the identities in different ways. It's a great way to demonstrate the fluidity of mathematical logic.

Establishing the Identities (STUDENT, TEACHER)

Establishing the Identities flipchart to pair with instruction. This is a very interactive flipchart so I strongly encourage you to read the page notes for suggestions on how to use it in your classroom.

PROJECT IDEAS

Glogster.com allows students (and you) to create an online poster. It's a great way to incorporate technology, its eco friendly and easier to grade than lugging around a lot of paper projects.

This is a student produced Glogster on trigonometry.

http://www.glogster.com/miachoi/trignometry-project-by-mia-choi/g-6lghcbo0pr0qqkkq4knjca0

Goanimate.com is a website which allows you to create animations for free. It is extremely user friendly and easy to maneuver. Two of the videos on this page were created using Goanimate.com. You can assign students different topics or allow them to choose a topic and create an animation about it.

Have students create their own flashcards and/or quizzes on www.quizlet.com. This is an especially good project for this unit because there are so many identities to remember.

There are tons of ways to assess students on this material outside of traditional quizzes and testing these are just a few.

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