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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