1.
Evaluate \(\tan\left(\frac{2\pi}{3}\right)\) using the definitino of tangent: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\text{.}\)
Section 6.2 Graphs of Other Trigonometric Functions
In this section, we’ll discuss the graphs of tangent, cosecant, and secant and their domains.
Subsection Textbook Reference
This relates to content in §8.2 of Algebra and Trigonometry 2e.
Exercises Preparation Exercises
Answer the following without using a calculator. These questions are intended to check the prerequisite skills needed to complete the rest of the lab.
2.
Evaluate \(\csc\left(-\frac{5\pi}{6}\right)\) using the definitino of cosecant: \(\csc(\theta) = \frac{\csc(\theta)}{\sin(\theta)}\text{.}\)
3.
Which of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are odd functions?
4.
What is the domain of the sine function? What is the domain of the cosine functions?
5.
Which values of \(\theta\) make it so that \(\sin(\theta)\\) is \(0\text{?}\) Hint: There are infinitely many answers.
Exercises Practice Exercises
Recall that for a right triangle with angle \(\theta\text{,}\) the "other trigonometric functions" can be found by the following definitions. Visit this interactive Desmos graph to see the six trigonometric lengths in action.
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\(\displaystyle \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
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\(\displaystyle \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\)
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\(\displaystyle \sec(\theta) = \frac{1}{\cos(\theta)}\)
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\(\displaystyle \csc(\theta) = \frac{1}{\sin(\theta)}\)
1.
(a)
The cosecant function, \(\csc(\theta) = \frac{1}{\sin(\theta)}\text{,}\) would be undefined when \(\sin(\\theta)=0\text{.}\) When is \(\sin(theta)=0\text{?}\) There are infinitely many answers: write your answers in a full sentence describing the pattern.
(b)
Write your answers to \(\sin(\theta)=0\) in set-builder notation.
(c)
Based on your answers to the above questions, what is the domain of the cosecant function? Write your answer in a sentence explaining the pattern.
2.
(a)
The secant function, \(\sec(\theta) = \frac{1}{\cos(\theta)}\text{,}\) would be undefined when \(\cos(\theta)=0\text{.}\) When is \(\cos(\theta)=0\text{?}\) There are infinitely many answers: write your answers in a full sentence describing the pattern.
(b)
Write your answers to \(\cos(\theta)=0\) in set-builder notation.
(c)
Based on your answers to the above questions, what is the domain of the secant function? Write your answer in a sentence explaining the pattern.
3.
(a)
The tangent function, \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\text{,}\) would be undefined when \(\cos(\theta)=0\text{.}\) This is the same problem that the secant function had. So, what is the domain of the tangent function? Write your answer in a sentence explaining the pattern.
(b)
The cotangent function, \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\text{,}\) would be undefined when \(\sin(\theta)=0\text{.}\) This is the same problem that the cosecant function had. So, what is the domain of the cotangent function? Write your answer in a sentence explaining the pattern.
4.
Match the "other trigonometric functions" with their graphs, shown in Figure 6.2.1, Figure 6.2.2, Figure 6.2.3, and Figure 6.2.4. Use the definitions of the functions and their domains to decide which goes with which. Check your answers with your favorite graphing program.
(a)
(b)
(c)
(d)
Exercises Exit Exercises
1.
Which trigonometric functions have a domain that is the set of all real numbers except integer multiples of \(\pi\text{?}\)
2.
Which trigonometric function is defined to be \(\dfrac{\cos(\theta)}{\sin(\theta)}\text{?}\)
3.
Which trigonometric functions have a domain that is the set of all real numbers except odd-integer multiples of \(\frac{\pi}{2}\)
4.
Shown in Figure 6.2.5 is a graph of a transformation of \(y = \tan(x)\text{.}\) What values of \(C\) and \(D\) create this transformation?
Reflection Reflection
1.
On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?
