Section 6.1 Inverse Trig Functions Section 6.2 Trig Equations I Section 6.3 Trig Equations II...

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Transcript of Section 6.1 Inverse Trig Functions Section 6.2 Trig Equations I Section 6.3 Trig Equations II...
Section 6.1 Inverse Trig Functions
Section 6.2 Trig Equations I
Section 6.3 Trig Equations II
Section 6.4 Equations
Chapter 6Inverse Trig Functions and Equations
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
Inverse Function
• The inverse function of the onetoone function f is defined as:
f 1 = {(y,x)  (x,y) belongs to f }
General Statements about Inverses
1. If the point (a,b) lies on the graph of the onetoone function ff, then the point (b,a) lies on the graph of ff 11..
2. The domain of ff is equal to the range of ff 11, and the range of ff is equal to the domain of ff 11.
3. For all x in the domain of ff , ff 11[(x)] = x, and all x in the domain of ff 11, ff [(x)]=x.
General Statements about Inverses
4. Because point (b, a) is a reflection of the point (a, b) across the line y=x, the graph of ff 11 is the reflection of the graph of ff across this line.
5. To find the equation that defines the inverse of a onetoone function ff , follow these steps:
1. Let y = ff (x)2. Interchange x and y in the equation3. Solve for y, and then write y= f f 11(x).
sin1x or arcsin x
y = sin1 x or y = arcsin x
means x=sin y, for y in [é/2, é/2]
cos1x or arccos x
y = cos1 x or y = arccos x
means x=cos y, for y in [0, é]
tan1x or arctan x
y = tan1 x or y = arctan x
means x=tan y, for y in (é/2, é/2)
sec1x or arcsec x
y = sec1 x or y = arcsec x
means y = cos1 (1/x),
where x is in (ë, 1] [1, ë)
csc1x or arccsc x
y = csc1 x or y = arccsc x
means y = sin1 (1/x),
where x is in (ë, 1] [1, ë)
cot1x or arccot x
y = cot1 x or y = arccot x means
y = tan1 (1/x),if x is in [0, ë)
or
y = tan1 (1/x) + é ,if x is in(ë, 0)
Section 6.2 Trig Equations I
• Solve equations by linear methods
• Solve equations by factoring
• Solve equations by quadratic formula
• Apply trig to music
Intersection of Graphs Method
• Graph each equation y=f(x) and y=g(x)
• Find the point(s) of intersection
xIntercept Method
• Graph y=f(x)
• Find the xIntercepts
• These are the solutions for f(x) = 0
Solving Trigonometric Equations Algebraically
1. Decide whether the equation is linear or quadratic (to determine the method)
2. If there is only on trig function, solve the equation for that function
3. 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.
4. If #3 doesn’t work try using an identity or square both sides.
5. If the equation is quadratic and not factorable use the quadratic formula.
Solving Trig Equations Graphically
For equations of the form f(x) = g(x):
1. Graph y=f(x) and y =g(x) over the required domain.
2. Find the xcoordinates of the points of intersection of the graphs.
Solving Trig Equations Graphically (cont)
For equations of the form f(x) = 0:
1. Graph y=f(x) over the required domain.
2. Find the xintercepts of the graphs.
Section 6.3 Trig Equations II
• Solve equations with halfangles
• Solve equations with multiple angles
• Apply trig to music
Section 6.4 Equations Involving Inverse Trig Functions
• Solve for x in terms of y using inverse functions
• Solve inverse trigonometric equations