3.08A - Basic Sinusoidal Functions

Trigonometry Lesson 3.08A Basic Sinusoidal Functions Cosine Sine Sinusoidal Functions are the relationship between the measure of angle, in radians, and the resulting value of the SINE or COSINE ratio. 𝑓 𝑥 = cos 𝑥 g 𝑥 = sin 𝑥 These functions are said to be “Periodic” as they repeat after a period of time. Key Terms • Cycle: One complete pattern is called a cycle. A cycle may begin at any point on the graph of the function. • Period (P): The period of a function is the horizontal length of one cycle. 𝑥2 − 𝑥1 • Frequency (F): The frequency is the reciprocal of the period. 1 𝐹= 𝑃 Example of a Periodic Function: 1 cycle • There are 2 full cycles of the periodic function above P = 𝑥2 − 𝑥1 P=8−0=8 1 cycle is 8 units long F= 1 𝑃 = 1 8 1 cycle happens every 8 units Key Terms (continued): • Sinusoidal Axis: the horizontal line that cuts the sinusoidal curve in half 𝑦= • 𝑚𝑎𝑥 + 𝑚𝑖𝑛. 2 Amplitude: half the distance between the max and the min 𝑚𝑎𝑥 − 𝑚𝑖𝑛 𝐴= 2 • Generalizing a solution: you have to give the answer for ONE cycle and then state that it is in every other cycle as well: 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ± 𝑃𝑛 where 𝑃 is the PERIOD of the Function and 𝑛 is the number of cycles 𝑛∈ℤ RULE OF THUMB: provide the solution(s) that is(are) the lowest positive Example of a Periodic Function’s Properties: 1 cycle Variation: of Properties: •Recap There are 2 full cycles of the periodic𝑓 ↗: function above −8, −6 ∪ −4, −1 dom 𝑓 = [−8, 8] P =ran 𝑥2𝑓−=𝑥1− 5 , 1 4 PIntercepts: =8−0=8 IV: 𝑓 0 = 0 23 17 1 cycle is 𝑧8 =units Zeros: {−8,long − 5 , −3, 0, 5 , 5, 8} ∪ 0, 2 ∪ [4, 7] 𝑓 ↘: 1−6, −4 1 ∪ −1, 0 ∪ 2, 4 ∪ 7,8 F= = Extremes: 𝑃 8 5 𝑚𝑖𝑛 = − 𝑚𝑎𝑥 = 1 4 1 cycle happens every 8 units Sign: 23 17 ∪ −3, ∪ 5, 8 5 5 23 17 𝑓 < 0: − , −3 ∪ ,5 𝑓 ≥ 0: −8, − • Lets look at the properties: 𝒇 𝒙 = 𝐜𝐨𝐬 𝒙 Period P = 𝑥2 − 𝑥1 P = 2𝜋 − 0 = 2𝜋 Frequency 1 1 F= = 𝑃 2𝜋 1 cycle Sinusoidal Axis 𝑦=0 • Let’s look at the properties 𝒈 𝒙 = 𝐬𝐢𝐧 𝒙 Period P = 𝑥2 − 𝑥1 P = 2𝜋 − 0 = 2𝜋 Frequency 1 1 F= = 𝑃 2𝜋 1 cycle Sinusoidal Axis 𝑦=0 Comparing Sine and Cosine Two Functions – One major Connection • If we graph both the sine and cosine functions on the same Cartesian Plane we can analyze them together. 𝒇 𝒙 = 𝐜𝐨𝐬 𝒙 Period 1 cycleFrequency P = 𝑥2 − 𝑥1 P = 2𝜋 − 0 = 2𝜋 1 1 F= = 𝑃 2𝜋 𝒈 𝒙 = 𝐬𝐢𝐧 𝒙 1 cycle Sinusoidal Axis 𝑦=0 𝒇 𝒙 = 𝐜𝐨𝐬 𝒙 • Both are curves oscillating between -1 and 1. min = −1 • 𝒈 𝒙 = 𝐬𝐢𝐧 𝒙 max = 1 Amplitude (A): Half the distance between the max and min. max − min 1 − (−1) 𝐴= = =1 2 2 𝒇 𝒙 = 𝐜𝐨𝐬 𝒙 𝒈 𝒙 = 𝐬𝐢𝐧 𝒙 • These two curves are identical save a horizontal translation of 2 units. • • This translation is called a PHASE SHIFT 𝜋 What this proves is the following: sin 𝑥 = cos 𝑥 − 𝜋 2 𝑂𝑅 cos 𝑥 = sin 𝑥 + 𝜋 2 𝒇 𝒙 = 𝐜𝐨𝐬 𝒙 𝒈 𝒙 = 𝐬𝐢𝐧 𝒙 • These two curves are identical save a horizontal translation of 2 units. • • This translation is called a PHASE SHIFT 𝜋 What this proves is the following: sin 𝑥 = cos 𝑥 − 𝜋 2 𝑂𝑅 cos 𝑥 = sin 𝑥 + 𝜋 2 Properties of Basic Functions cos 𝑥 sin 𝑥 ℝ ℝ −1, 1 −1, 1 1 0 𝑥 = 𝜋𝑛 ; where 𝑛 ∈ ℤ 𝑓↗ 𝜋 𝑥= + 𝜋𝑛 ; where 𝑛 ∈ ℤ 2 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 𝑓↘ 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 Domain Range Intercepts: 𝑓(0) 𝑓 𝑥 =0 Variation: Extremes: max min Sign 𝑓≥0 𝑓≤0 Vertex 1 −1 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 (0,1) 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 1 −1 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 (0,0) Properties of Basic Functions cos 𝑥 sin 𝑥 ℝ ℝ −1, 1 −1, 1 1 0 𝑥 = 𝜋𝑛 ; where 𝑛 ∈ ℤ 𝑓↗ 𝜋 𝑥= + 𝜋𝑛 ; where 𝑛 ∈ ℤ 2 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 𝑓↘ 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 Domain Range Intercepts: 𝑓(0) 𝑓 𝑥 =0 Variation: Extremes: max min Sign 𝑓≥0 𝑓≤0 Vertex 1 −1 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 (0,1) 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 1 −1 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 (0,0) Properties of Basic Functions cos 𝑥 sin 𝑥 ℝ ℝ −1, 1 −1, 1 1 0 𝑥 = 𝜋𝑛 ; where 𝑛 ∈ ℤ 𝑓↗ 𝜋 𝑥= + 𝜋𝑛 ; where 𝑛 ∈ ℤ 2 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 𝑓↘ 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 Domain Range Intercepts: 𝑓(0) 𝑓 𝑥 =0 Variation: Extremes: max min Sign 𝑓≥0 𝑓≤0 Vertex 1 −1 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 (0,1) 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 1 −1 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 (0,0) Properties of Basic Functions cos 𝑥 sin 𝑥 ℝ ℝ −1, 1 −1, 1 1 0 𝑥 = 𝜋𝑛 ; where 𝑛 ∈ ℤ 𝑓↗ 𝜋 𝑥= + 𝜋𝑛 ; where 𝑛 ∈ ℤ 2 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 𝑓↘ 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 Domain Range Intercepts: 𝑓(0) 𝑓 𝑥 =0 Variation: Extremes: max min Sign 𝑓≥0 𝑓≤0 Vertex 1 −1 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 (0,1) 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 1 −1 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 (0,0) Properties of Basic Functions Domain Range Intercepts: 𝑓(0) 𝑓 𝑥 =0 Variation: 𝑥= ℝ ℝ −1, 1 −1, 1 1 0 𝜋 3𝜋 ± 2𝜋𝑛, ± 2𝜋𝑛 ; 𝑛 ∈ ℤ 2 2 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 𝑓↘ 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 min 𝑓≥0 𝑓≤0 Vertex sin 𝑥 𝑓↗ Extremes: max Sign cos 𝑥 1 −1 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 (0,1) 𝑥 = 0 ± 2𝜋𝑛, 𝜋 ± 2𝜋𝑛 ; 𝑛 ∈ ℤ 𝜋 𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 1 − −1 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 (0,0) Properties of Basic Functions Domain Range Intercepts: 𝑓(0) 𝑓 𝑥 =0 Variation: 𝑥= ℝ ℝ −1, 1 −1, 1 1 0 𝜋 3𝜋 ± 2𝜋𝑛, ± 2𝜋𝑛 ; 𝑛 ∈ ℤ 2 2 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 𝑓↘ 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 min 𝑓≥0 𝑓≤0 Vertex sin 𝑥 𝑓↗ Extremes: max Sign cos 𝑥 1 −1 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 (0,1) 𝑥 = 0 ± 2𝜋𝑛, 𝜋 ± 2𝜋𝑛 ; 𝑛 ∈ ℤ 𝜋 𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 1 − −1 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 (0,0) Properties of Basic Functions Domain Range Intercepts: 𝑓(0) 𝑓 𝑥 =0 Variation: 𝑥= ℝ ℝ −1, 1 −1, 1 1 0 𝜋 3𝜋 ± 2𝜋𝑛, ± 2𝜋𝑛 ; 𝑛 ∈ ℤ 2 2 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 𝑓↘ 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 min 𝑓≥0 𝑓≤0 Vertex sin 𝑥 𝑓↗ Extremes: max Sign cos 𝑥 1 −1 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 (0,1) 𝑥 = 0 ± 2𝜋𝑛, 𝜋 ± 2𝜋𝑛 ; 𝑛 ∈ ℤ 𝜋 𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 1 − −1 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 (0,0) Properties of Basic Functions Domain Range Intercepts: 𝑓(0) 𝑓 𝑥 =0 Variation: 𝑥= ℝ ℝ −1, 1 −1, 1 1 0 𝜋 3𝜋 ± 2𝜋𝑛, ± 2𝜋𝑛 ; 𝑛 ∈ ℤ 2 2 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 𝑓↘ 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 min 𝑓≥0 𝑓≤0 Vertex sin 𝑥 𝑓↗ Extremes: max Sign cos 𝑥 1 −1 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 (0,1) 𝑥 = 0 ± 2𝜋𝑛, 𝜋 ± 2𝜋𝑛 ; 𝑛 ∈ ℤ 𝜋 𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 1 − −1 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 (0,0) Properties of Basic Functions Domain Range Intercepts: 𝑓(0) 𝑓 𝑥 =0 Variation: 𝑥= ℝ ℝ −1, 1 −1, 1 1 0 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 ; 𝑛 ∈ ℤ 2 2 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 𝑓↘ 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 min 𝑓≥0 𝑓≤0 Vertex sin 𝑥 𝑓↗ Extremes: max Sign cos 𝑥 1 −1 𝜋 𝜋 − + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 (0,1) 𝑥 = 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 ; 𝑛 ∈ ℤ 𝜋 𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 𝜋 3𝜋 + 2𝜋𝑛, + 2𝜋𝑛 2 2 1 − −1 0 + 2𝜋𝑛, 𝜋 + 2𝜋𝑛 𝜋 + 2𝜋𝑛, 2𝜋 + 2𝜋𝑛 (0,0) Visual Ordered Pairs 𝒇 𝒙 = 𝐜𝐨𝐬 𝒙 𝟎, 𝟏 𝝅 ,𝟎 𝟐 𝝅, −𝟏 𝟑𝝅 ,𝟎 𝟐 (𝟐𝝅, 𝟏) 𝒈 𝒙 = 𝐬𝐢𝐧 𝒙 𝟎, 𝟎 𝝅 ,𝟏 𝟐 𝝅, 𝟎 𝟑𝝅 , −𝟏 𝟐 (𝟐𝝅, 𝟎) Summary of Terms and Formulas: • • Cycle: the piece of the function that repeats (the stamp) • Frequency: the reciprocal of the period 𝐹 = 𝑃 • Sinusoidal Axis: the horizontal line that cuts the sinusoidal curve in half Period: the distance between the endpoints of ONE cycle 1 𝑚𝑎𝑥 + 𝑚𝑖𝑛. 𝑦= 2 • Amplitude: half the distance between the max and the min 𝑚𝑎𝑥 − 𝑚𝑖𝑛 𝐴= 2 • Generalizing Statements: 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ± 𝑃𝑛; 𝑛 ∈ ℤ