Math 170 - Cooley Pre-Calculus OCC
Section 5.4 – The Other Trigonometric Functions and Their Graphs
Definition – Tangent, Cotangent, Secant and Cosecant Functions
If ∝ is an angle in standard position and is the point of Intersection of the terminal side and the unit circle, we define the tangent, cotangent, secant, and cosecant functions as
Identities from the Definitions
If ∝ is any angle or real number
provided no denominator is zero.
The Graph of y = tanx
The fundamental cycle of is the piece of the graph bolded over the interval .
Properties of the Tangent Function y = tanx
1. The domain is the set of all real numbers, except odd multiples of .
2. The range is the set of all real numbers.
3. The tangent function is an odd function, as the symmetry of the graph with respect to the origin indicates.
4. The tangent function is periodic, with period p.
5. The x-intercepts are …, –2p, –p, 0, p, 2p, 3p,…; the y-intercept is 0.
6. Vertical asymptotes occur at
The Graph of y = cotx
The fundamental cycle of is the piece of the graph bolded over the interval .
Properties of the Cotangent Function y = cotx
1. The domain is the set of all real numbers, except multiples of .
2. The range is the set of all real numbers.
3. The cotangent function is an odd function, as the symmetry of the graph with respect to the origin indicates.
4. The cotangent function is periodic, with period p.
5. The x-intercepts are …, ; there is no y-intercept.
6. Vertical asymptotes occur at x = –2p, –p, 0, p, 2p, 3p,…
The Graph of y = cscx
The fundamental cycle of is the piece of the graph bolded over the interval .
Properties of the Cosecant Function y = cscx
1. The domain is the set of all real numbers, except multiples of .
2. The range is the set of all real numbers greater than or equal to 1 and all real numbers less than or equal to –1.
3. The cosecant function is an odd function, as the symmetry of the graph with respect to the origin indicates.
4. The cosecant function is periodic, with period 2p.
5. There are no x-intercepts and no y-intercepts.
6. Vertical asymptotes occur at x = –2p, –p, 0, p, 2p, 3p,…
The Graph of y = secx
The fundamental cycle of is the piece of the graph bolded over the interval .
Properties of the Secant Function y = secx
1. The domain is the set of all real numbers, except odd multiples of .
2. The range is the set of all real numbers greater than or equal to 1 and all real numbers less than or equal to –1.
3. The secant function is an even function, as the symmetry of the graph with respect to the y-axis indicates.
4. The secant function is periodic, with period 2p.
5. There are no x-intercepts. The y-intercept is 1.
6. Vertical asymptotes occur at
J Exercises:
Find the exact value of each function. Do not use a calculator.
1) 2) 3) 4) 5)
Find the approximate value of each expression. Round answers.to four decimal places.
6) 7) 8) 9)
J Exercises:
Determine the period and phase shift for each function.
10) 11)
12) 13)
Graph each function. Sketch at least one cycle of the graph.
14)
J Exercises:
Graph each function. Sketch at least one cycle of the graph.
15)
16)
17)
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