Section 6.1 Inverse Trig Functions

1. Chapter 6Inverse Trig Functions and Equations Section 6.1 Inverse Trig Functions Section 6.2 Trig Equations I Section 6.3 Trig Equations II Section 6.4 Equations

2. Section 6.1 Inverse Trig Functions • Know and use the inverse sine • Know and use the inverse cosine • Know and use the inverse tangent • Know and use the inverse secant • Know and use the inverse cosecant • Know and use the inverse cotangent • Know and use inverse function values

3. Inverse Function • The inverse function of the one-to-one function f is defined as: f -1 = {(y,x) | (x,y) belongs to f }

4. General Statements about Inverses • If the point (a,b) lies on the graph of the one-to-one function f, then the point (b,a) lies on the graph of f-1. • The domain of f is equal to the range of f-1, and the range of f is equal to the domain of f-1. • For all x in the domain of f, f-1[(x)] = x, and all x in the domain of f-1, f[(x)]=x.

5. General Statements about Inverses • Because point (b, a) is a reflection of the point (a, b) across the line y=x, the graph of f-1 is the reflection of the graph of facross this line. • To find the equation that defines the inverse of a one-to-one function f, follow these steps: • Let y = f(x) • Interchange x and y in the equation • Solve for y, and then write y= f-1(x).

6. sin-1x or arcsin x y = sin-1 x or y = arcsin x means x=sin y, for y in [-é/2, é/2]

7. cos-1x or arccos x y = cos-1 x or y = arccos x means x=cos y, for y in [0, é]

8. tan-1x or arctan x y = tan-1 x or y = arctan x means x=tan y, for y in (-é/2, é/2)

9. sec-1x or arcsec x y = sec-1 x or y = arcsec x means y = cos-1 (1/x), where x is in (-ë, -1]  [1,ë)

10. csc-1x or arccsc x y = csc-1 x or y = arccsc x means y = sin-1 (1/x), where x is in (-ë, -1]  [1,ë)

11. cot-1x or arccot x y = cot-1 x or y = arccot x means y = tan-1 (1/x),if x is in [0,ë) or y = tan-1 (1/x) + é ,if x is in(-ë, 0)

12. Section 6.2 Trig Equations I • Solve equations by linear methods • Solve equations by factoring • Solve equations by quadratic formula • Apply trig to music

13. Intersection of Graphs Method • Graph each equation y=f(x) and y=g(x) • Find the point(s) of intersection

14. x-Intercept Method • Graph y=f(x) • Find the x-Intercepts • These are the solutions for f(x) = 0

15. Solving Trigonometric Equations Algebraically • Decide whether the equation is linear or quadratic (to determine the method) • If there is only on trig function, solve the equation for that function • If there are more than one, rearrange the equation so that one side equals 0. Try to factor and set each factor = 0 to solve. • If #3 doesn’t work try using an identity or square both sides. • If the equation is quadratic and not factorable use the quadratic formula.

16. Solving Trig Equations Graphically For equations of the form f(x) = g(x): • Graph y=f(x) and y =g(x) over the required domain. • Find the x-coordinates of the points of intersection of the graphs.

17. Solving Trig Equations Graphically (cont) For equations of the form f(x) = 0: • Graph y=f(x) over the required domain. • Find the x-intercepts of the graphs.

18. Section 6.3 Trig Equations II • Solve equations with half-angles • Solve equations with multiple angles • Apply trig to music

21. Section 6.4 Equations Involving Inverse Trig Functions • Solve for x in terms of y using inverse functions • Solve inverse trigonometric equations