5.1 Identities
The reciprocal identity for
cos(x)
What is
1/sec (x)?
The proof to the identity: cos^2\theta /(1-cos^2theta)
What is
cos^2theta-1?
What is the reciprocal of:
cotx
1/cotx or tanx?
All solutions to the equation in the interval [0,2\pi) . cos(x)=-sqrt3/2
what is
x=(2pi)/3,(4pi)/3
Solve for the missing trigonometric values. sin x=1/2, cos x=sqrt3/2
tanx=sqrt3/3 , cscx=2,secx=(2sqrt3)/3, cotx=sqrt3
The value of the following expression:
sin^2 (x)+cos^2(x).
What is
1?
The proof to the identity:
sinx/cosx+cosx/sinx = secx\cdot cscx
sin^2x/(cosxsinx)+cos^2x/(cosx sinx)
1/(sinx cosx)
secx tanx
What is the reciprocal of:
sinx/cosx
cosx/sinx or cotx
All solutions to the equation in the interval [0,2\pi) . 2sintheta-1=0
What is
theta=pi/3, (2pi)/3
Find the missing angle. Write the general solution.
sin(2x)=-sqrt3/2
sin(t)=-sqrt3/2
t=2x
t=(4pi)/3, (5pi)/3
x=(2pi)/3+npi,n\in \mathbb{Z}
x=(5pi)/6+npi,n\in \mathbb{Z}
The simplified version of:
1+tan^2(x).
What is
sec^2(x)?
The proof to the identity:
sin\phi/(cos\phi )cot\phisin\phi=sin\phi
tanphicotphisinphi
1sinphi
sinphi
Rewrite using an identity:
(3sinx)/cos^2x = 3tanxsecx
3 sinx/cosx 1/cosx
3tanxsecx
All solutions to the equation in the interval [0,2\pi) . cot(x)=sqrt3
x=pi/6,(7pi)/6
What is the distance between the points P=(-15,23) and Q=(-7,8)
sqrt((-7-(-15))^2+(8-23)^2)
sqrt((8)^2+(15)^2)
sqrt(289)
17
The simplified version of:
sin(x)/cos(x).
What is
tan(x)?
The proof to the identity:
sin^2\theta/(cos^2\theta)+cos^2\theta/cos^2theta=sec^2\theta
tan^2theta+1
sec^2theta
Rewrite using an identity: sin\theta/sectheta+cos\theta/csctheta
sin theta cos theta/1 +costhetasintheta/1
sinthetacostheta+sinthetacostheta
2sinthetacostheta
All solutions to the equation in the interval [0,2\pi) . 3csc^2x-4=0
csc^2x=4/3
cscx=+-2/sqrt3
sinx=+-sqrt3/2
x=pi/3, (2pi)/3, (4pi)/3, (5pi)/3
Write three ways to write tan^2(\theta )
1/cot^2theta
sec^2theta-1
sin^2theta/cos^2theta
The simplified version of:
cos^2(\theta)-1.
What is
-sin^2(\theta)?
The proof to the identity:
(1+cosx)/(1-sinx)=(secx+tanx)(secx+1)
((1+cosx)(1+sinx))/((1-sinx)(1+sinx))
(1+cosx+sinx+cosxsinx)/(cos^2x)
sec^2x+secx+tanxsecx+tanx
secx(secx+1)+tanx(secx+1)
(secx+tanx)(secx+1)
Rewrite using an identity:
(5sin^2theta-11sintheta+2)/(5sintheta-1)=sintheta-2
((sintheta-10/5)(sin theta-1/5))/(5sintheta-1)
((sintheta-2)(5sin theta-1))/(5sintheta-1)
sintheta-2
All solutions to the equation in the interval [0,2\pi) . sec^2x-secx=2
sec^2x-secx-2=0
(secx-2)(secx+1)=0
secx=2 or secx=-1
cosx=1/2 or cosx=-1
x=pi/3, (5pi)/3, pi
The distance between the top of my head and the top of my shadow If I am 72 inches tall and I cast a 52.32 inch shadow. Assume that the angle between me and the ground is a right angle.
Let d be the distance between the tip of the shadow and the top of my head. Then
d^2=(72)^2+(52.32)^2
d=sqrt((72)^2+(52.32)^2)
d=sqrt(7921.3824)
d=89