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.

The purpose of this discussion is for students to make connections between features of the graphs of the secant, cosecant, and cotangent functions and the graphs of other periodic functions they have studied.

Invite students to do a Gallery Walk of the displays. As they inspect each other’s graphs, they should consider these questions: 

• “How are the graphs similar to and different from each other?”
• “How are the graphs similar to and different from the graphs of the sine, cosine, and tangent functions?”

After the Gallery Walk, invite 3–5 students to share any similarities and differences they noticed among the graphs. Record their responses for all to see. Some things students may notice are:

• Cotangent, cosecant, and secant all have a period of .
• Cotangent and cosecant have asymptotes at 0, , , while secant has asymptotes at and .
• Cosecant and cotangent have the same asymptotes, while secant and tangent have the same asymptotes.
• Only sine and cosine have maximums and minimums.
• Secant and cosecant never have a value between -1 and 1.

Students may not notice all of these, but if no students mention features like the maximum, minimum, and period of the functions, ask them to discuss these features.

The purpose of this activity is to provide students with an additional opportunity to examine the secant, cosecant, and cotangent functions, including the features of the graphs of those functions.

In this partner activity, students take turns matching graphs and expressions representing the same functions. As students analyze different representations, they practice reasoning quantitatively and abstractly (MP2). In making connections across representations, they practice looking for and making use of structure (MP7).

The purpose of this discussion is for students to make connections between the trigonometric ratios and functions of sine and cosecant, cosine and secant, and tangent and cotangent. 

Once all groups have completed the Card Sort, discuss: 

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

Invite students to arrange the cards with graphs in pairs in this way:

•  with   (Card B with Card D)
•  with   (Card A with Card E)
•  with   (Card C with Card F)

Next, ask students: "Look at Cards B and D. How are cosecant and sine related?" (Cosecant is the reciprocal of sine.) Display this graph and identity:

Then ask, "How do you see this relationship on the graphs of sine and cosecant?" (Wherever  , cosecant is undefined. Cosecant never takes a value of 0, which makes sense since there are no undefined values for sine. Sine and cosecant are both 1 at the same values of .)

Invite students to look at Cards A and E. Display this graph and identity:

Then ask, "Why does  have asymptotes at  and ?" (Since secant is the reciprocal of cosine, wherever  , secant is undefined. Secant never takes a value of 0, which makes sense since there are no undefined values for cosine.) 

Invite students to consider Cards C and F. Display this graph and identity:

Then ask students: "How could you decide whether a graph is of the cotangent or tangent function?" (The graph of  is always increasing, while the graph of  is always decreasing, except at the asymptotes. The graph of  crosses the origin, while the graph of  has an asymptote at .)