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?

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

x

1

2

3

4

–1

–2

–3

–4

y

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

x

1

–1

y

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

x

1

2

3

4

5

6

–1

y

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