CHAPTER 7 TRIGONOMETRIC GRAPHS 7 1 GRAPHS OF

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 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

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)

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

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

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. 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)

7. 1: GRAPHS OF SIN/COS/TAN