Trigonometry Assignment
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Date Due
Part 1: Understanding why Identities in Trigonometry are important.
1. What does the text say is a key purpose of using trig identities?
The text say that using trigonometric identities is essential in proving trigonometric relationships which helps one to become more creative towards solving problems. This is attributable to the fact that by learning how to establish trigonometric relationships one becomes a better as well as a confident problem solver which is an important aspect of a successful life (page 642).
2. In your own words, what does the book say about solving conditional equations vs. verifying identities?
In solving a conditional equation one has to work with both sides of the equation concurrently through addition, subtraction, multiplication or division both sides using similar expression. However, verification of an identity is done through manipulation of each side of the equation in an independent manner from the other side (page 644).
3. When attempting to verify that a given equation is an identity, am I allowed to multiply or add the same quantity to both sides of the equation? Explain your reason(s).
When verifying whether a certain equation is an identity, it is allowed to multiply the same quantity to both sides of the equation; but it is not allowed to add the same quantity to both sides of the equation. Multiplication is allowed when multiplying the numerator and denominator by the same factor to verify an identity probably when there are no fractions to combine as well as when none of the sides of the equation which tend to be more complicated compared to the other one (page 647). However, verification of an identity can’t be done by adding the same quantity to both sides of the equation since by doing so one has already assumed that the given statement is true (page 644).
Part 2: Identifying the most useful techniques when verifying trig identities
1. Various techniques Blitzer uses to verify that the given trig equations are identities.
There are various techniques used to verify whether trigonometric equations are identities depending on their nature:
a. Changing sines and consines to verify an identity
b. Using factoring to verify an identity
c. Using two techniques to verify an identity
i. Separating a single-term quotient into two terms
ii. Changing to sin and consines
d. Combining fractional expressions to verify an identity
e. Multiplying the numerator and denominator by the same factor to verify an identity
f. Working with both sides separately to verify an identity
2. Often used techniques to verify an identity
Blitzer identifies that working with both sides separately to verify an identity is the technique which is often used, and it is used in Example 8.
3. Did any of the Fundamental Trig identities in this section define any way to change one angle into a different one?
Yes. Where angle is changed to sines and consines
Now, turn to page 671 and the table of Principal Trig Identities. What is “principally” being manipulated in this set of formulas?
The angles in terms of tangent, sine and cosine
Part 3: Verifying some trig identities
Exercise 3
tan(-x) cos〖x=-sinx 〗
sin〖(A/2)=±√((1-cosA)/2)〗 tan(A/2)=csc〖A-cotA 〗
COS A/2=±√((1+cosA)/2) cot(A/2)=csc〖A+cotA 〗
Exercise 7
sec〖x-sec〖x 〖sin〗^2 〗 〗 x=cosx
sin〖x=(e^ix-e^(-ix))/2i〗
cos〖x=(e^ix+e^(-ix))/2〗
tan〖x=(e^ix-e^(-ix))/(i(e^ix+e^(-ix)))〗
Exercise 11
csc〖θ-sin〖θ=cot〖θ cosθ 〗 〗 〗
sec(θ)-tan(θ) sin(θ)=1/(cos(θ))-(sin(θ))/(cos(θ)).(sin(θ))/1
=1/(cos(θ))-(〖sin〗^2 (θ))/(cos(θ))
=(〖cos〗^2 (θ))/(cos(θ))
=cos(θ)
Exercise 16
〖cos〗^2 θ(1+〖tan〗^2 θ)=1
tan(θ)+cot(θ)=(sin(θ))/(cos(θ))+(cos(θ))/(sin(θ))
=(〖sin〗^2 (θ))/(cos(θ) sin〖(θ)〗 )+(〖cos〗^2 (θ))/(cos(θ)sin(θ))
=(〖sin〗^2 (θ)+〖cos〗^2 (θ))/(cos(θ)sin(θ))
=1/(cos(θ)sin(θ))
=sec(θ)csc(θ)
Exercise 21
(〖tan〗^2 t)/sect =sec〖t-cost 〗
3tan x/2+3=0
=3tan x/2+3-3=-3
=3tan x/2=-3
=tan x/2=-1
tan x/2=tan(π-π/4)
=tan(3π/4)
x/2=3π/4
x/2=3π/4+nπ
x=3π/2+2nπ
Exercise 26
sint/tant +cost/cott =sin〖t+cost 〗
〖sec〗^4 (θ)-〖sec〗^2 (θ)=〖tan〗^4 (θ)+2〖tan〗^2 (θ)+1-〖sec〗^2 (θ)
=〖tan〗^4 (θ)+2〖tan〗^2 (θ)+(-〖tan〗^2 (θ))
=〖tan〗^4 (θ)+〖tan〗^2 (θ)
Exercise 31
cosx/(1-sinx )+(1-sinx)/cosx =2secx
(2-tan(θ))/(2 csc(θ)-sec(θ))=(2-(sin(θ))/(cos(θ)))/(2 1/(si(θ))-1/(cos(θ)))
=((2 cos(θ)-sin(θ))/(cos(θ)))/((2 cos(θ)-sin(θ))/(sin(θ)cos(θ)))
=((2 cos(θ)-sin(θ))/(cos(θ)))((sin(θ)cos(θ))/(2 cos(θ)-sin(θ)))
=sin(θ)
Exercise 36
csc〖x-secx 〗/csc〖x+secx 〗 =cot〖x-1〗/cot〖x+1〗
2〖sec〗^2 A=1/(1-sinA )+1/(1+sinA )
RHS=1/(1-sinA )+1/(1+sinA )
=(1+sin〖A+1-sinA 〗)/((1-sin〖A)(1+sin〖A)〗 〗 )
=2/(1-〖sin〗^2 A)
2/(〖cos〗^2 A)
=2〖sec〗^2 A
Exercise 41
tan〖2θ+cot2θ 〗/csc2θ =sec2θ
=tan(θ)+cot(θ)=(sin(θ))/(cos(θ))+(cos(θ))/(sin(θ))
=(〖sin〗^2 (θ))/(cos(θ) sin(θ))+(〖cos〗^2 (θ))/(cos(θ)sin(θ))
=(〖sin〗^2 (θ)+〖cos〗^2 (θ))/(cos(θ)sin(θ))
=1/(cos(θ)sin(θ))
=sec(θ)csc(θ)
Part 4: The Most Important Trig Identity in the Universe.
1. First, look at Figure 1a and 1b. Describe in words what Blitzer did between 6.1a and 6.1b.
Blitzer has just rotated the triangle so that point P falls on the x-axis at point (1, 0), a rotation which has changed the coordinates of both points P and Q.
2. Why is the Unit Circle, as opposed to a larger or smaller circle centered elsewhere, preferred?
The Unit Circle is used to make sure that the lengths and angles of the triangle fitted inside it remains equal despite the changes in coordinates of points P and Q.
3. On page 654, Blitzer has TWO separate formulas for the length of PQ. How did he build them?
The two separate equations are built using the distance formula and then equating the two expressions of PQ obtained before and after rotation of the triangle which results to variation in the coordinates of both points as shown in Figure 6.1(a) and Figure 6.1(b).
4. What Fundamental Identity plays a key role in simplifying each formula for the length of PQ?
The Pythagorean Identity
5. What are the final steps Blitzer takes to derive the Most Important Trig Identity in the Universe?
The final steps taken to derive the trigonometric identity is equating the two expressions of PQ followed by simplification of the equation which begins by squaring both sides to remove radicals.
6. State that identity here. Use a large font and bold weight to write it.
cos(α-β)=cos α cosβ+sin〖α sinβ 〗
Part 5
Exercise 2
cos(〖120〗^0-〖45〗^0)
=cos〖120〗^0 cos〖〖45〗^0 〗+sin〖〖120〗^0 sin〖〖45〗^0 〗 〗
=1/2 1/√2+√3/2 1/√2
=(1+3)/(2√2)
Exercise 3
cos(3π/4-π/6)
=2 〖sin〗^2 2 x+cos〖2(2x)〗
=2〖sin〗^2 2x+〖cos〗^2 2x-〖sin〗^2 2x
=〖sin〗^2 2x+〖cos〗^2 2x
Exercise 9
(cos(∝-β))/cos〖∝sinβ 〗 =tan〖∝〖+bcot〗β 〗
tan(x-π/2)=(sin(x-2))/(cos(x-π/2))
=sin〖x cos〖(π/2)-sin〖(π/2) cosx 〗 〗 〗/(cos〖x cos〖(π/2〗 〗)+sin〖x sin〖(π/2)〗 〗 )
=(0 sin〖x-1 cosx 〗)/(0 cos〖x+1 sinx 〗 )
=(-cosx)/sinx =-cotx
Exercise 15
sin〖〖105〗^0 〗
sin〖〖105〗^0 〗=sin(1/2 〖52.5〗^0 )=+√((1+cos〖〖52.5〗^0 〗)/2)
=+√((1+√3/2)/2=+√((2+√3)/4))
=+√((2+√3)/2)
Exercise 24
tan(5π/3-π/4)
=(2 tanx)/(1-〖tan〗^2 x)=tanx
=2tan〖x=tan〖x(1-〖tan〗^2 x)〗 〗
=2tan〖x=tan〖x-〖tan〗^3 〗 〗 x
=〖tan〗^3 x+tan〖x=0〗
=tan〖x(〖tan〗^2+1)=0〗
Exercise 30
sin 7π/12 cos π/12-cos 7π/12 sin π/30
=2〖sin〗^2 x/2+cosx
=2(√((1-cosx)/2))^2+cosx
=2 ((1-cosx)/2)+cosx
Exercise 33
sin(x-π/2)=cosx
sin2x/(1-cos2x)=(2 sin〖x cosx 〗)/(1-(1-2〖sin〗^2 x)
=(2 sin〖x cosx 〗)/(2 sin〖x sinx 〗 )
=cosx/sinx =cotx
Exercise 36
cos(π-x)=-cosx
=2〖 cos〗^2 x/2-cosx
=2(√((1+cosx)/2))^2-cosx
=2((1-cosx)/2)-cosx
=1+cos〖x-cosx 〗
=1
Exercise 39
sin(∝+β)+sin(∝-β)=2sin〖∝cosβ 〗
=cos(〖30〗^0+ x
=cos〖〖30〗^0 cos〖 x sin〖〖30〗^0 sinx 〗 〗 〗
=√3/2 cos〖x-1/2 sinx 〗
=(√3 cos〖x-sinx 〗)/2
Exercise 57
sin〖x=2 sin〖x cosx 〗 〗
=2.(-a)/√(a^2+b^2 ).(-b)/√(a^2+b^2 )
=2ab/(a^2+b^2 )
Exercise 60
tan〖2x=(2 tanx)/(1-〖tan〗^2 x)=(2 a/b)/(1-(a/b)^2 )〗
(2a/b)/((b^2-a^2)/b^2 )=2a/b.b^2/(b^2-a^2 )
=2ab/(b^2-a^2 )
Exercise 62
cos(1/2)=√((1-cos(x))/2)
=√((1-(-9/41))/2)
√(50/82)=√(25/41)
=5/√41=(5√41)/41
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