Day 4: 10- 4&5 Domain and Range/Inverse Trig Functions

SWBAT: (1) Evaluate inverse trig functions

(2) Identify the domain and range of trig functions.

Do Now: Find in degrees, if 0o≤ < 360

o: sin = ½

Concept 1: Domain and Range of the trig functions.

Sine and Cosine What numbers are we allowed to put into f(x) = sin(x) _____________________ What numbers are we allowed to put into f(x) = cos(x) _____________________ Therefore, the domain of sine and cosine is ___________. The range of both sine and cosine are different. As we get our y-coordinates from the unit circle, as we rotate around, what is the largest and smallest values we get for the y’s? ______________________________ Therefore, the range of sine and cosine is ___________.

Tangent and Secant Tangent is different because of its nature. tan(x) = ______________ sec(x) = ______________ Therefore, the domain of tangent and Secant is _____________________________________. Therefore, the range of tangent is ___________.

To find its range were going to use a little inequality we know from cos(x). We know the range of the cos(x) is between [-1,1] we are going to split it and do some algebra on it to figure out what sec(x) must be.

1)cos(0 x or 0)cos(1 x

Therefore, the range of secant is _________________________________________.

Cotangent and Cosecant cot(x) = ______________ csc(x) = ______________ Therefore, the domain of tangent and Secant is ___________________________________. Cotangent is similar to tangent, Therefore, the range of cotangent is

Cosecant is similar to secant, It’s range can be found exactly the same as we did in sec(x), Therefore, the range of cosecant is

Summary:

Concept 2: Domain and Range of Inverse trig functions The inverse trig functions are __________________________________________ To construct inverse functions, we must have a property that our original functions are

Is Sin 1-1 or not? ___________ In order to make the inverse of sin we must restrict our domain in the original.

Creating the inverse sine function Sine Inverse sine or _____________________ Domain -__________________ Domain - __________________________ Range - __________________ Range - ___________________________

Is Cos 1-1 or not? ___________ In order to make the inverse of cos we must restrict our domain in the original.

Creating the inverse cosine function Cosine Inverse cosine or _____________________ Domain -__________________ Domain - __________________________ Range - __________________ Range - ___________________________

Creating the inverse tangent function

Is Tan 1-1 or not? ____________ Then we must restrict it’s domain too! Tangent Inverse tangent or ____________________ Domain -__________________ Domain - __________________________ Range - __________________ Range - ___________________________

Summary:

Find each value of : a) in degrees b) in radians

Concept 3: Calculating the Degree Measure of inverse Trig Functions For inverse trig evaluations, be sure to use the restricted range values! Part One: Find the value of .

1) = Arc cos 2

3 2) = Arc tan -1 3) =Arc sin

2

1

4) = Arc sec ( √ ) 5) = Arc cot 3 6) = Arc csc ( √

)

Part Two: Find the value of

7) Find to the nearest degree: = Arc cos (-.6). 8) Find to the nearest degree: = Arc sin (

).

Part Three: Find the value of These problems involve 2 steps of evaluation, so do the inner one first. For inverse trig evaluations, be sure to use the restricted range values!

9) Find the exact value: sin (Arc tan 1) 10) Find the exact value: sec

2

2sinArc

10) Find the exact value: csc (Arc cos

) 11) Find the exact value: cot (Arc tan √ )

SUMMARY

Example 1:

Exit Ticket 1)

2)

10- 4&5 Domain and Range/Inverse Trig Functions : Homework

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Answer Key