Finding the angle when given the function value

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Objective

SWBAT find all special angles that have a given angle.

Big Idea

How many angles have the same trigonometric value.

Bell Work

5 minutes

Now that students have had time to work on finding the value of trigonometric functions, I want to change things up a bit. Today students will be given the value of a trigonometric function and find all angles that have that value.

I begin by having students consider this question:

Find all angles between 0 and 2*pi that have a sine value of 1/2.

I let students work for a few minutes and then share their results. We discuss how students determined the answer. I'll ask a question like, "Why are there 2 answers?"

Find the angle

20 minutes

After discussing the Bell Work I give students a value and ask students to find all trigonometric functions including the angle that has that value. Students work on this for a minute or two and then start putting their answers on the board. I do not limit the values that the angle can be to see if students consider rotating multiple times around the circle.

As students share their answers I watch for answers that are more than one revolution or that are negative. If students have not considered this I will put a few of these on the board for students to verify if they are correct or incorrect. I will put at least one that is wrong just to make sure students are checking my answers.

How are the angles related to the ones on the board?

When students say that they are 360 degrees or 2pi radians more than the given angle. I now ask students if they can think of a way to write all angles? I show students the idea of 2npi or 360n along with 180n or npi.  How many answers could we have?

I now become more specific and give students a trigonometric equation such as sin t= 1.  Students work to find the answer. We do several of these before I give students problems with a limited domain.

Individual Practice

10 minutes

Once I feel that students are understanding how to find the angle I give student the Special Angles Worksheet 2.  I help students read through the directions. One issue that students have is how to determine if the answer needs to be in radians or degrees. I show students how reading the directions and noticing the limits on the domain will help them know which way to write the angles. On this worksheet the domain limits are in radians so the answers should be in radians.

As students work today I help students that are still struggling. Many students will want me to just verify that the answers are correct. Once they know they are doing the problems correctly the students are willing to work in small groups on the problems.

Closure

5 minutes

As class ends today I ask students to complete and exit slip to answer this prompt.

When we are given the value and the function such as sin t=1/2, we can have an infinite number of solutions or just 1 or 2. What should you find to help you determine the number of solutions for a problem like sin t=1/2?

Many times students do not read the directions to a problem or consider the situation to a real world problem when determining an answer. Students need to realize they need to identify the domain to determine the number of solutions a problem will have.