Unit 4 Trigonometric Formulas

Unit 4 Trigonometric Formulas ● Trigonometric Identities 𝑠𝑖𝑛𝜃 1 ○ 𝑡𝑎𝑛𝜃 = 𝑐𝑜𝑠𝜃 = 𝑐𝑜𝑡𝜃 ○ 𝑐𝑜𝑡𝜃 = 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 1 = 𝑡𝑎𝑛𝜃 1 ○ 𝑠𝑒𝑐𝜃 = 𝑐𝑜𝑠𝜃 ○ 𝑐𝑠𝑐𝜃 = ○ 1 𝑠𝑖𝑛𝜃 Even ■ cos(-θ) = cosθ ■ sec(-θ) = secθ ○ Odd ■ sin(-θ) = -sinθ ■ csc(-θ) = -cscθ ■ tan(-θ) = -tanθ ■ cot(-θ) = -cotθ ○ Pythagorean ■ cos2θ +sin2θ =1 ■ sec2θ = 1+tan2θ ■ csc2θ =1+cot2θ ● Sum and Difference Formulas ○ sin(α +β)=sinαcosβ +cosαsinβ ○ cos(α +b)=cosαcosβ - sinαsinβ ○ tan(α+β=tanα+tanβ ● Half-Angle Formulas 1−𝑐𝑜𝑠𝜃 𝜃 ○ 𝑠𝑖𝑛 2 = ±√ 2 𝜃 1+𝑐𝑜𝑠𝜃 ○ 𝑐𝑜𝑠 2 = ±√ 𝜃 ○ 𝑡𝑎𝑛 2 = 2 1−𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜃 = 1+𝑐𝑜𝑠𝜃 ● Double Angle Formulas ○ 𝑠𝑖𝑛(2𝜃) = 2𝑠𝑖𝑛θcosθ ○ 𝑐𝑜𝑠(2𝜃) = 2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 = 2𝑐𝑜𝑠 2 𝜃 − 1 = 1 − 2𝑠𝑖𝑛2 2𝑡𝑎𝑛𝜃 ○ 𝑡𝑎𝑛(2𝜃) = 1−𝑡𝑎𝑛2 𝜃 Circles and Angles ● Vocab ○ Angles : 2 rays with a common vertex ■ Vertex : Point where sides/rays meet, point of rotation ■ Initial side : Beginning of angle/rotation ■ Terminal Side : End of angle/rotation ○ Positive vs Negative Angles : Positive angle (counter clockwise), Negative angle (clockwise) ○ Standard position of angles ■ Vertex : (0,0) ■ Initial Side : Positive X-axis ● Degrees, Minutes, and seconds (D’, M’, S”) ○ DMS to degree Calculator Instructions ■ [2nd][APPS][1] to insert the degree symbol ■ [2nd][APPS][2] to insert the symbol for minutes ■ [ALPHA][+] to insert the symbol for seconds ○ Degree to DMS Calculator Instructions ■ Press [MODE], use the arrow keys to highlight DEGREE, and then press [ENTER] ■ Enter the degree measure ■ Press [2nd][APPS][4][ENTER] to convert the degrees to DMS ○ 1 degree = 60 minutes and 1 minute = 60 seconds ○ Example: Convert 60° 25’ 32” into decimal degrees 25° 32° ■ 60 ∘ + 60° + 3600 = 60.425° ● Radians ○ Radian is a ratio of 2 distances as angle graws arc length gets wider ○ 𝜃= 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑟𝑎𝑑𝑖𝑢𝑠 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ ○ 1 radian = 𝑟𝑎𝑑𝑖𝑢𝑠 ○ Circumference = 2πr ○ Full Circle 2𝜋𝑟 ■ 𝜃= 𝑟 = 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 360° ○ Half Circle ■ π radians = 180° ○ Converting radians to degrees 𝜋 ■ 290° ⋅ 180 = 29𝜋 18 ≈ 5.06 ○ Converting degrees to radians ■ 2.5 ⋅ ○ Arc length 180° 𝜋 = 450 𝜋 = 143.239° ■ s = rθ ● s = arc length ● r = radius ● θ = angle in radians ○ Area of sector ■ A = ½ r2 θ ● A = area ● r = radius ● θ = angle in radians ○ Circular Motion 𝑠 ■ 𝑣 = 𝑡(linear speed) 𝜃 ■ 𝜔 = 𝑟 (angular speed) ■ 𝑣 = 𝑟𝜔(relationship between linear and angular speed) Unit Circle and Trigonometric Functions ● Circle: x2 + y2 = r2 ● Unit Circle: x2 + y2 = 1 ○ r = 1 in the unit circle ● Center: (0,0) ● Trigonometric functions: ○ 𝑠𝑖𝑛𝜃 = 𝑦 𝑟 = 𝑜𝑝𝑝 ℎ𝑦𝑝 𝑥 𝑎𝑑𝑗 𝑦 𝑜𝑝𝑝 ○ 𝑐𝑜𝑠𝜃 = 𝑟 = ℎ𝑦𝑝 ○ 𝑡𝑎𝑛𝜃 = 𝑥 = 𝑎𝑑𝑗 𝑟 ℎ𝑦𝑝 𝑟 ℎ𝑦𝑝 𝑥 𝑎𝑑𝑗 ○ 𝑐𝑠𝑐𝜃 = 𝑦 = 𝑜𝑝𝑝 ○ 𝑠𝑒𝑐𝜃 = 𝑥 = 𝑎𝑑𝑗 ○ 𝑐𝑜𝑡𝜃 = 𝑦 = 𝑜𝑝𝑝 ● Quadrantal Angles: Angles that lie on an axis ○ Value of the 6 trig functions at θ = 0 radians ■ 𝑠𝑖𝑛𝜃 = 𝑦 𝑟 =0 𝑥 ■ 𝑐𝑜𝑠𝜃 = 𝑟 = 1 𝑦 ■ 𝑡𝑎𝑛𝜃 = 𝑥 = 0 𝑟 ■ 𝑐𝑠𝑐𝜃 = 𝑦 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑟 ■ 𝑠𝑒𝑐𝜃 = 𝑥 = 1 𝑥 ■ 𝑐𝑜𝑡𝜃 = 𝑦 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝜋 ● Reference Angle: an angle that is directly between (0,90) or (0, ) 2 ● Periodic Function: Function that repeats outputs after a period of time ○ 𝑓(𝜃 + 𝑝) = 𝑓(𝜃) ■ p = period ● Period: length of time(input) until a function repeats ● Periods of each trig function ○ Sine: 2π ○ Cosine: 2π ○ Tangent: π ○ Cosecant: 2π ○ Secant: 2π ○ Cotangent: π ● Quadrants trig identities are positive in ○ I: All ○ II: sin and csc ○ III: tan and cot ○ IV: cos and sec ■ Pneumonic Device: All Students Take Calculus ● How to find the values for the other 5 functions, when given 1 function: 1. Identify the quadrant and determine the sign of x,y, and r 2. Assign values to 2 of the following x, y, and r 3. Use 𝑥 2 + 𝑦 2 = 𝑟 2 to find the remaining piece and remember the sign 4. Use definitions and identities to find the remaining 4 functions Graphing Trig Functions ● 𝑦 = 𝑠𝑖𝑛𝑥 ○ Amplitude(A) = 1 ○ Period(T) = 2π ○ Domain: (-∞,∞) ○ Range: [-1,1] ● 𝑦 = 𝑐𝑜𝑠𝑥 ○ Amplitude(A) = 1 ○ Period(T) = 2π ○ Domain: (-∞,∞) ○ Range: [-1,1] ● Transformations 𝐶 ○ 𝐴𝑠𝑖𝑛(𝐵𝑥 + 𝐶) + 𝐷 𝑜𝑟 𝑦 = 𝐴𝑠𝑖𝑛(𝐵 (𝑥 + 𝐵)) + 𝐷 ■ Amplitude: |𝐴| ■ Period: 2𝜋 3 ■ C: Horizontal Shift ■ D: Vertical Shift ■ Interval Width: Period x ¼ ● 𝑦 = 𝑠𝑒𝑐𝑥 ○ Amplitude(A) = A ○ Period(T) = 2𝜋 𝐵 ○ Domain: {x| x ≠ odd 𝜋 multiples of 2} ○ Range: (-∞,-1] ∪ [1,∞) ● 𝑦 = 𝑐𝑠𝑐𝑥 ○ Amplitude(A) = A ○ Period(T) = 2𝜋 𝐵 ○ Domain: {x| x ≠ integer multiples of 𝜋} ○ Range: (-∞,-1] ∪ [1,∞) ● 𝑦 = 𝑡𝑎𝑛𝑥 ○ Amplitude(A) = A 𝜋 ○ Period(T) = 𝐵 𝜋 ○ Domain: {x| x ≠ odd multiples of 2} ○ Range: (-∞,∞) ● Phase Shifts ○ Bx + C = 0 ■ X = phase shift Inverse Trig ● 𝑦 = 𝑠𝑖𝑛−1 𝑥 ○ Domain: [-1,1] −𝜋 𝜋 ○ Range: [ 2 , 2] ● 𝑦 = 𝑐𝑜𝑠 −1 𝑥 ○ Domain: [-1,1] ○ Range: [0, 𝜋] ● 𝑦 = 𝑡𝑎𝑛−1 𝑥 ○ Domain: (-∞,∞) −𝜋 𝜋 ○ Range: [ 2 , 2] ● 𝑦 = 𝑐𝑠𝑐 −1 𝑥 ○ Domain:(-∞,-1]∪[1,∞) 𝜋 𝜋 ○ Range: [0, 2 ] ∪ [ 2 , 𝜋] ● 𝑦 = 𝑐𝑜𝑡 −1 𝑥 ○ Domain: [-∞,∞] ○ Range: (0, 𝜋) Solving Trigonometric Equations 𝜋 ● 𝜃 = 3 + 2𝜋𝑘 𝑜𝑟 𝜃 = 5𝜋 + 2𝜋𝑘 3 ● Isolate trig function ● Example: 2𝑠𝑖𝑛𝜃 + √3 = 0 ○ 2𝑠𝑖𝑛𝜃 2 = ○ 𝑠𝑖𝑛𝜃 = −√3 2 −√3 2 ○ Answers: ■ 𝜃= 4𝜋 + 2𝜋𝑘 3 𝑜𝑟 𝜃 = 5𝜋 + 2𝜋𝑘 3