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Chapter 7: Trigonometric Graphs 7.1: Graphs of the Sine, Cosine, and Tangent Functions

Chapter 7: Trigonometric Graphs 7.1: Graphs of the Sine, Cosine, and Tangent Functions. Essential Question: How do you graph a transformation of the sine, cosine, and tangent graphs?. 7.1: Graphs of Sin/Cos/Tan.

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Chapter 7: Trigonometric Graphs 7.1: Graphs of the Sine, Cosine, and Tangent Functions

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  1. Chapter 7: Trigonometric Graphs7.1: Graphs of the Sine, Cosine, and Tangent Functions Essential Question: How do you graph a transformation of the sine, cosine, and tangent graphs?

  2. 7.1: Graphs of Sin/Cos/Tan • We’ve actually already touched upon this concept, back in 6-4, when I focused on the wave approach instead of the unit circle approach. • Since the waves repeat at on a cycle of 2π, we can add 2π repeatedly to find similar angles. • Example 1: State all values for t where sin t = -1 • The sin curve starts at 0, peaks (1) at π/2, and dips lowest (-1) at 3π/2 before returning to 0 at 2π. • So we say t = 3π/2 + 2πk, where k is any integer • State all values for t where cos t = ½ At t = π/3, cos t = ½. It also = ½ at -π/3, which is 5π/3 t = π/3 + 2πk or t = 5π/3 + 2πk, where k is any integer

  3. 7.1: Graphs of Sin/Cos/Tan • Using the calculator to solve the same type of problem • Example 3: State all values of t where tan t = -1 • It’s easiest if we start in degree mode • Use tan-1(-1) [the inverse tangent function] to get -45° • Convert -45° to radians • -45° • 2π/360° = -π/4 radians • Answer: t = -π/4 + πk, where k is any integer • Remember, tan repeats every π radians • Finding the second angle (sin/cos only) • sin t = sin( - t) • cos t = cos(-t)

  4. 7.1: Graphs of Sin/Cos/Tan • Basic Transformations • Vertical Stretches • Just like with regular functions, vertical stretches occur on the outside of the parent function. • See the graph to the right • Red = cos t • Blue = 4 cos t

  5. 7.1: Graphs of Sin/Cos/Tan • Basic Transformations • Reflections and Vertical Stretches • Just like with regular functions, vertical reflections occur on the outside of the parent function. • See the graph below • Red = sin t • Blue = -½ sin t

  6. 7.1: Graphs of Sin/Cos/Tan • Basic Transformations • Vertical Shifts • Just like with regular functions, vertical shifts occur on the outside (generally, after) of the parent function. • See the graph to the right • Red = tan t • Blue = tan t + 5

  7. 7.1: Graphs of Sin/Cos/Tan • Assignment • Page 483 – 484 • Draw a sketch of sin t from -2π to 2π (two cycles) (pg 475) • Draw a sketch of cos t from -2π to 2π (two cycles) (pg 477) • Draw a sketch of tan t from –π/2 to 3π/2 (two cycles) (pg 480) • It’ll help you tremendously with problems 7 – 19 (and the future quiz) • Problems 7 – 43, odd • For problems 35 – 43, just list the transformation (ignore the directions about stating the domain and range)

  8. 7.1: Graphs of Sin/Cos/Tan

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