1. Trigonometric Functions
Chapter 6
Trigonometric Functions

2. Section 5 Graphs of Tangent, Cotangent, Cosecant, and Secant Functions

3. DAY 1

4. As before, we are only going to be graphing one period or cycle of the graph X is still 𝜃 from unit circle and y = csc 𝜃 or y = sec 𝜃

5. Understanding the basic y = csc x
Understanding the basic y = csc x *remember: this is the reciprocal of sin x* X = 𝜃 y = csc 𝜃 0 undefined 𝜋/2 1 𝜋 undefined 3𝜋/2 -1 2𝜋 undefined *undefined on a graph = vertical asymptotes (approaches but doesn’t cross)

6. Main Characteristics of csc: Comparison: x-int of sin = V. A
Main Characteristics of csc: Comparison: x-int of sin = V.A. of csc max/min of sine = vertices of parabolas for csc (parabolas should always point away from x-axis) Center: ½ way between vertices Period: 2𝜋 Amplitude: distance from vertices to center Domain: {x| x ≠ whole # multiples of 𝜋} Range: {y| y ≥ 1 or y ≤ -1}

7. Understanding the basic y = sec x
Understanding the basic y = sec x *remember: this is the reciprocal of cos x* X = 𝜃 y = sec 𝜃 0 1 𝜋/2 undefined 𝜋 -1 3𝜋/2 undefined 2𝜋 1 *undefined on a graph = vertical asymptotes (approaches but doesn’t cross)

8. Main Characteristics of sec: Comparison: x-int of cos = V. A
Main Characteristics of sec: Comparison: x-int of cos = V.A. of sec max/min of cos = vertices of parabolas for sec (parabolas should always point away from x-axis) Center: ½ way between vertices Period: 2𝜋 Amplitude: distance from vertices to center Domain: {x| x ≠ odd multiples of 𝜋/2} Range: {y| y ≥ 1 or y ≤ -1}

9. Example: Graph y = -3 csc(x + 𝜋/4)

10. Example: Graph y = sec(x - 𝜋/2)

11. DAY 2

12. Understanding the basic y = tan x. remember: tan = sin/cos = y/x
Understanding the basic y = tan x *remember: tan = sin/cos = y/x* X = 𝜃 y = tan 𝜃 0 0 𝜋/4 1 𝜋/2 undefined 3𝜋/4 -1 𝜋 0 5𝜋/4 1 3𝜋/2 undefined 7𝜋/4 -1 2𝜋 0 *undefined on a graph = vertical asymptotes (approaches but doesn’t cross)

13. Main Characteristics of tan: V. A
Main Characteristics of tan: V.A.  where cos x = 0  odd multiples of 𝜋/2 x-intercepts  sin x = 0  whole # multiples of 𝜋 Period: 𝜋 Domain: {x| x ≠ odd multiples of 𝜋/2}  same as sec x Range: {y| all reals}

14. Understanding the basic y = cot x. remember: cot = cos/sin = s/y
Understanding the basic y = cot x *remember: cot = cos/sin = s/y* X = 𝜃 y = cot 𝜃 0 undefined 𝜋/4 1 𝜋/2 0 3𝜋/4 -1 𝜋 undefined 5𝜋/4 1 3𝜋/2 0 7𝜋/4 -1 2𝜋 undefined *undefined on a graph = vertical asymptotes (approaches but doesn’t cross)

15. Main Characteristics of cot: V. A
Main Characteristics of cot: V.A.  where sin x = 0  whole # multiples of 𝜋 x-intercepts  cos x = 0  odd multiples of 𝜋/2 Period: 𝜋 Domain: {x| x ≠ whole # multiples of 𝜋}  same as csc x Range: {y| all reals}

16. Example: Graph y = cot (x + 𝜋/4) + 1

17. Example: Graph y = 2 tanx – 1

18. EXIT SLIP