Composite Function



TRIGONOMETRY REVIEW

Circles

• Angles measured using degrees, radians, or revolutions

• One full revolution around a circle is equal to 360( and 2( radians

• Degree or radian measurements are equivalent over full revolutions

o E.g. 90( is equivalent to 450(, 90( plus 360(k where k is an integer

• Arc length

o Arc length for whole circle (or circumference) is 2(r

o Arc length is equal to (r

o Example: Arc Length of circle of radius 2 at (=(/2

o [pic] arc length = (/2 (2) =(

• Area of a sector:

o area of whole circle is (r2

o the area of a sector with angle ( is [pic]

Trigonometric Identities

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• The phrase SOH – CAH – TOA is a very useful when finding sin (, cos (, tan ( when given the lengths of sides of a triangle.

Sin=Opposite/Hypotenuse

Cos=Adjacent/Hypotenuse

Tan=Opposite/Adjacent

• [pic]

• [pic]

• [pic]

Proofs

• sin(A+B)

[pic]

[pic]

• cos(A+B)

[pic][pic]

[pic]Sorry Arup I was having trouble understanding what to type for the proof.

• tan(A+B)

[pic]

• sin2(

[pic]

• cos2[pic]

[pic]

• tan2[pic]

[pic]

Finding Ratios with out finding the angle.

• When given a ratio, for example [pic] it is possible to find other ratios such as cos[pic], tan[pic], sin2[pic], cos2[pic][pic], etc.

• To find the other trig functions with a single [pic], just use the Pythagorean theorem. Using the example [pic]:

[pic] 52-22=x2 or 25-4=21=x2

So cos[pic]= [pic], tan[pic]= [pic], etc.

• To find trig functions with more than 1 [pic], manipulate the trigonometric identities.

o E.g. give that [pic] to sin2[pic] use sin2[pic]=2sin[pic]cos[pic]

2sin[pic]cos[pic]=[pic]

The Nature of trigonometric graphs

• Sinx

o Has period of 2(

o Is always continuous

• Cosx

o Has period of 2(

o Is always continuous

o Similar to graph of sinx but offset by (/2 to the left

• Tanx

o Has period of (

o Is undefined where x=(/2 + k(

• K is an integer

Composite Function

f(x) = a sin(b(x+c)) + d

• a changes the amplitude of the graph.

• b changes the period of the graph, condensing or expanding it

• c moves the graph left or right

• d moves the graph up or down.

Inverse Functions

• arcsin x: has a range of (-[pic],[pic]) [pic]

has a domain of [-1,1]

• arccos x: has a range of (-[pic],[pic]) [pic]

has a domain of [-1,1]

• arctan x: has a range of [-[pic],[pic]] [pic]

has a domain of (-[pic],[pic])

Rules and Formulas for Triangles

• A = [pic]bh

• Hero’s Formula: A = [pic] where s = [pic]

• Cosine Rule: [pic]

• The Sine Rule: [pic]

• Alternate Area: A = [pic]absinC

Half Angle Identities:

• We know that [pic]. From this it can be determined that [pic] and [pic].

• If we replace 2A by [pic] and use t to denote [pic] we have:

o [pic]

o [pic]

o [pic]

More Identities

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

• [pic]

Practice Problems

1) A chord AB divides a circle of radius 2m into two segments. If AB subtends an angle of [pic] at the center of the circle, find the area of the minor segment.

2) Find the sine, cosine, and tangent of [pic].

3) Find the general solution of [pic].

4) Solve [pic] for angles in the range [pic].

5) Without using tables or a calculator, evaluate:

(a) [pic]

(b) [pic]

(c)[pic]

6) Prove that:

[pic]

7) Express [pic] in terms of [pic].

Solutions

1.

[pic] [pic]

Area of sector AOB = [pic]

Area of [pic]AOB = [pic]

Area of Minor Segment (shaded in) = (2.094 – 1.732) [pic] = .362[pic]

2. [pic]

[pic]

[pic]

[pic]

3. [pic] because the tangent function has a period of [pic],

[pic]

4.

[pic]

5.

(a)[pic]

(b)[pic]

(c)[pic]

6.

[pic]

7. Using [pic] where [pic] gives

[pic]

and [pic]

Hence [pic]

[pic]

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