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Chapter 4 Trigonometric Functions Chapter 4: ● ● ● ● ● ● ● 4.1-Right Triangle Trigonometry 4.2-Degrees and Radians 4.3-Trigonometric Functions on the Unit Circle 4.4-Graphing Sine and Cosine Functions 4.5-Graphing Other Trigonometric Functions 4.6-Inverse Trigonometric Functions 4.7-The Law of Sines and the Law of Cosines 4.1 Right Triangle Trigonometry 4.1 Concepts ● ● ● ● ● Learning Trig Ratios Learning the Special Right Triangles Finding sides of Right Triangles with the Trig ratios Finding angles of Right Triangles with the Trig ratios Solving word problems with Trig 4.1 Vocab! ● ● ● ● ● Trigonometric Ratios= see below Sine, Cosine, Tangent, Cosecant, Secant, Cotangent Special Right Triangles= 30-60-90, 45-45-90 Inverse Trig Functions= see second dot Angles of Depression and Elevation= 4.1 Formulas ● opp/hyp, adj/hyp, opp/adj, hyp/opp, hyp/adj, adj,opp ● 30-60-90 ● 45-45-90 4.2 Degrees and Radians 4.2 Concepts ● Converting radians to degrees ● Converting degrees to radians ● Use the measures of angles to solve word problems 4.2 vocabulary ● vertex- a common endpoint for two noncollinear rays ● initial side- the starting position of a ray is the initial side of an angle ● terminal side-the rays position after rotation ● standard position- when there is an angle at the origin and its initial side along the positive x-axis 4.2 Formulas ● ● ● ● ● ● ● ● radians/180 =radians 180/radians=degrees degrees+360n=degrees possible radians+2n=radians possible s=r(theta) A=½r^2(theta) v=s/t w=(theta)/t 4.3 Trigonometric Functions on the Unit Circle 4.3 Concepts ● ● ● ● ● ● Evaluate trigonometric functions given a point Evaluate trigonometric functions of quadrantal angles Find reference angles Use reference angles to find trigonometric functions Use one trigonometric function to find others Find trigonometric values using the unit circle (the one you know by heart) 4.3 Vocabulary quadrantal angle-an angle in standard position that has a terminal side that lies on one of the coordinate axes reference angle- the acute angle formed by the terminal side of an angle in standard position and the x-axis unit circle- circle of radius 1 centered at the origin of a coordinate system circular function- a trigonometric function defined as the function of the real number system using the unit circle periodic function- a function with values that repeat at regular intervals period- for a function y=f(t), the smallest positive number c for which f(t+c)= f(t) 4.3 Formulas ● ● ● ● ● ● ● r=√(x2 +y2) ≠0 tan(theta)= y/x cot(theta)= x/y sin(theta)=y/r csc(theta)=r/y cos(theta)= x/r sec(theta) r/x 4.4 Graphing Sine and Cosine Functions a hilarious joke…. "sine and cosine had to choose their way of travel. Which way did they go?" ... "They went off on a tangent!" 4.4 Concepts ● ● Graphing Sine Functions (with transformation) ○ vertical dilation (change in amplitude) ○ reflections over x - axis (add negative the equation ex. -3cos(x)) ○ horizontal transformation (shifting left or right) ○ vertical transformations (shifting up or down) Graphing Cosine Functions (with transformation) ○ vertical dilation (change in amplitude) ○ reflections over x - axis (add negative the equation ex. -3cos(x)) ○ horizontal transformation (shifting left or right) ○ vertical transformations (shifting up or down) Parent Graphs SINE PARENT GRAPH y=sin(x) COSINE PARENT GRAPH y=cos(x) 4.4 Vocabulary sinusoid: any transformation of a sine graph amplitude: half the distance between the max and min midline: the line that becomes the reference line or equilibrium point when the graph is shifted up or down frequency: the number of cycles in one interval phase shift: function shifts to the left or right period: time it take of the function to make one cycle vertical shift: graph goes through vertical transformation (up or down) 4.4 Formulas f(x) = (a)sin(b(x-c)) +d a ----- amplitude (vertical stretch/shrink) b ----- change in period c ----- horizontal shift d ----- vertical shift How to Find the Period: 2(π)/ b f(x) = (a)cos(b(x-c)) +d a ----- amplitude (vertical stretch/shrink) b ----- change in period c ----- horizontal shift d ----- vertical shift How to Find the Period: 2(π)/ b Example Problems f(x) = -3 sin (π/2(x+2π) - 1 f(x) = 4 cos (2(x- π/4) + 1 Solution: Solution: 4-5 Graphing Other Trigonometric Functions (Tangent and Reciprocals) Topics Topics: ● Graphing tangent functions ● Graphing cotangent functions ● Graphing secant functions ● Graphing cosecant functions Formulas Formulas: ● y= a tan (bx+c) ● y= a cot (bx+c) ● P= pi/|b| (for tangent and cotangent) ● P= 2pi/|b| (for secant and cosecant) ● y= a sec (bx+c)+d ● y= a csc (bx+c)+d Terms period of a tangent function: the distance between any two consecutive vertical asymptotes 4.6 Inverse Trigonometry Functions Concepts ● ● y=sin-1 X is the same as y=arcsin X This means the angle between the restricted domains and ranges Formulas ● ● ● Inverse Sine of X: The angle between -Pi/2 and Pi/2 Y=sin-1 X Domain: [-1,1] Range: [-Pi/2, Pi/2] ● ● ● Inverse Cosine of X: The angle between 0 and Pi Y=cos-1 X Domain: [-1, 1] Range: [0, Pi] ● ● ● Inverse of Tangent of X: The angle between -Pi/2 and Pi/2 Y=tan-1 X Domain: (-infinity, infinity) Range: (-Pi/2, Pi/2) 4.7 Law of Sines and Cosines!!!! Concepts ● ● ● ● Using Law of Sines to Solve a Triangle Using Law of Cosines to Solve a Triangle The Ambiguous Case Finding the Area of a Triangle Vocab ● ● ● ● ● Law of Sines= a formula Law of Cosines= another formula The Ambiguous Case= SSA Heron's Formula= the area Oblique Triangles= not right Formulas ● ● ● ● a2 = b2 + c2 - 2bc Cos(A) s sqrt(s-a)(s-b)(s-c) s = 1/2(a+b+c) sinA/a= sinB/b= sinC/c The AMBIGUOUS Case A good way to set it up (thanks Olivia) THE RULES Why you should study... The End