5.4 More Trigonometric Graphs

---

5.4.1 Learning Objectives

By the end of this section, you should be able to:

• graph tangent and cotangent functions;
• graph secant and cosecant functions;
• identify periods, asymptotes, domains, and ranges;
• graph transformations of these functions; and
• use Desmos to verify key features.

---

5.4.2 Tangent and Cotangent Graphs

The tangent function has period \(\pi\), zeros at \(x=k\pi\), and vertical asymptotes at \(x=\pi/2+k\pi\). Cotangent also has period \(\pi\), with zeros at \(x=\pi/2+k\pi\) and asymptotes at \(x=k\pi\).

Example 5.13 Graphing a Tangent Transformation

Find the period and asymptotes of \(y=\tan(2x)\).

Solution

The period is \(\pi/2\). Asymptotes occur when \(2x=\pi/2+k\pi\):

\[ \boxed{\text{Period }\pi/2;\quad x=\pi/4+k\pi/2}. \]

Exercise 5.13 Find the period of \(y=\cot(3x)\).

Show answer

Answer: \(\boxed{\pi/3}\)

---

5.4.3 Secant and Cosecant Graphs

Secant is \(1/\cos x\), so its vertical asymptotes occur where cosine is zero. Cosecant is \(1/\sin x\), so its asymptotes occur where sine is zero. Both have range

\[ (-\infty,-1]\cup[1,\infty). \]

Their graphs can be sketched by first drawing the related cosine or sine guide curve.

Example 5.14 Finding Secant Features

State the period, asymptotes, and range of \(y=2\sec x-1\).

Solution

The period is \(2\pi\), and asymptotes remain at \(x=\pi/2+k\pi\). Transforming the basic ranges gives

\[ \boxed{\text{Range }(-\infty,-3]\cup[1,\infty)}. \]

Exercise 5.14 State the vertical asymptotes of \(y=\csc(2x)\).

Show answer

Answer: \(\boxed{x=k\pi/2,\quad k\in\mathbb Z}\)

---

5.4.4 Transformations and Desmos

For tangent and cotangent, the period of \(A\tan(B(x-C))+D\) or \(A\cot(B(x-C))+D\) is \(\pi/|B|\). For secant and cosecant, the period is \(2\pi/|B|\). Horizontal and vertical shifts move asymptotes as well as graph branches.

Example 5.15 Describing a Cosecant Transformation

Describe \(y=-3\csc(x-\pi/2)+2\).

Solution

The graph shifts right \(\pi/2\), reflects across the \(x\)-axis, stretches vertically by \(3\), and shifts up \(2\). Its period is \(2\pi\).

\[ \boxed{\text{Right }\pi/2,\quad \text{reflect},\quad \text{stretch }3,\quad \text{up }2} \]

Exercise 5.15 Find the period and phase shift of \(y=\tan(4(x+\pi/8))\).

Show answer

Answer: \(\boxed{\text{Period }\pi/4,\quad \text{shift left }\pi/8}\)

---

5.4.5 Conceptual Takeaways

• Tangent and cotangent repeat every \(\pi\).
• Secant and cosecant inherit asymptotes from cosine and sine zeros.
• Reciprocal graphs never take values strictly between \(-1\) and \(1\).
• Transformations move both branches and asymptotes.
• Related sine and cosine curves help construct reciprocal graphs.

---

5.4.6 Skills You Should Be Able to Do

• Graph tangent and cotangent.
• Graph secant and cosecant using guide curves.
• Calculate periods and locate asymptotes.
• Determine domains and ranges.
• Verify transformed graphs with Desmos.

---

5.4.7 Practice Problems with Solutions

1. State the period of \(y=\tan(5x)\).

   Show Solution

   \[ \boxed{\pi/5} \]

2. State the vertical asymptotes of \(y=\tan x\).

   Show Solution

   \[ \boxed{x=\pi/2+k\pi,\quad k\in\mathbb Z} \]

3. State the period and asymptotes of \(y=\cot(2x)\).

   Show Solution

   The period is \(\pi/2\). Since \(\sin(2x)=0\) when \(2x=k\pi\),

   \[ \boxed{\text{Period }\pi/2;\quad x=k\pi/2}. \]

4. State the domain and range of \(y=\sec x\).

   Show Solution

   \[ \boxed{\text{Domain }x\ne\pi/2+k\pi;\quad \text{range }(-\infty,-1]\cup[1,\infty)}. \]

5. Find the range of \(y=4\csc x+1\).

   Show Solution

   From \(\csc x\le-1\) or \(\csc x\ge1\), multiplying by \(4\) and adding \(1\) gives

   \[ \boxed{(-\infty,-3]\cup[5,\infty)}. \]

6. Use Desmos to graph \(y=\sec(2x)\). State its period and asymptotes.

   Show Solution

   The period is \(2\pi/2=\pi\). Asymptotes occur when \(\cos(2x)=0\):

   \[ \boxed{\text{Period }\pi;\quad x=\pi/4+k\pi/2}. \]

---