Unit 6, Lesson 13
Some New Ratios
Integrated Math 3
13.1
Activity
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is for students to see the geometric relationships of sine, cosine, and tangent on a unit circle so they can build a more intuitive understanding of the reciprocal ratios—cosecant, secant, and cotangent. This Warm-up prompts students to make sense of a problem before solving it, by familiarizing themselves with a context and the mathematics that might be involved (MP1). Students will continue to make sense of this image as they see additional ratios and lengths for new trigonometric ratios added in future activities.
Launch
Arrange students in groups of 2. Display the image. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
Student Response
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Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the question of how the lengths change as the angle changes does not come up during the conversation, ask students to discuss this idea. Here are some questions to elicit student thinking on this idea:
- “How do the lengths labeled ‘sine,’ ‘cosine,’ and ‘tangent’ change as increases from 0 to ?” (Sine starts at 0, since the height of point is 0, and increases to 1. Cosine does the opposite, starting at 1 and decreasing to 0. Tangent also starts at 0, but it gets larger and larger. When the angle is , the tangent line is parallel to the -axis, so it isn't possible to find the length of the line segment between them.)
- “How does this correspond to the functions we have graphed of sine, cosine, and tangent?” (The values for sine and cosine between 0 and follow that same pattern. For the graph of the tangent function, this corresponds to the asymptote when the input is .)
13.2
Activity
Instructional Routines
NoneMaterials
To Copy (from Blackline Masters)
Some Additional Lines Handout
To Gather
Tools for creating a visual displayActivity Narrative
The purpose of this activity is to give students a chance to graph more trigonometric functions, specifically cosecant, secant, and cotangent. Students examine the triangle ratios, trigonometric identities, and geometric structure of cosecant, secant, and cotangent and leverage these structures to create graphs of the functions (MP7). Students will create a display that can be used as a reference, as they compare the additional trigonometric functions.
There is an optional blackline master that students may use to create their displays.
Launch
Arrange students in groups of 4. Assign each group to secant, cosecant, or cotangent.
Display this image and the question: "How does the length of the line you were assigned change as the angle increases from 0 to ?"
Tell students that just like the sine, cosine, and tangent, the secant, cosecant, and cotangent can also be thought of in several ways. They can be represented as a length on a diagram like this (of a point on the unit circle) or as a ratio between sides of a right triangle.
Display these ratios:
- cosecant:
- secant:
- cotangent:
Ask students to match these ratios with their identities: , , and (Cotangent, cosecant, and secant, respectively. The ratios are the reciprocals of the ratios for tangent, sine, and cosine.)
Keep the image, ratios, and identities displayed throughout the activity for student reference. Tell students that we will use these representations to help us build the cosecant, secant, and cotangent functions.
There is an optional blackline master available for students to use for the second and third questions as students create a visual display of their assigned functions.
Student Lesson Summary
We can use the sine, cosine, and tangent functions to talk about some new trigonometric functions: cosecant, secant, and cotangent.
Cosecant is the reciprocal function of sine. This means that . On a right triangle, we can identify the trigonometric ratio . We can use this information to create a graph of . The graph will have an asymptote any time . We also know that any time , any time , and has the same period as . Here is the graph of :
Secant is the reciprocal function of cosine. This means that . On a right triangle, we can identify the trigonometric ratio . We can use this information to create a graph of . The graph will have an asymptote any time . We also know that any time , any time , and has the same period as . Here is the graph of :
Cotangent is the reciprocal function of tangent. This means that . On a right triangle, we can identify the trigonometric ratio . We can use this information to create a graph of . The the graph will have an asymptote any time , and it will have a value of 0 any time has an asymptote. We also know that any time , any time , and has the same period as . Here is the graph of :