Fundamental Identities
Find the reciprocal identity:
csc u
1/sin u
Verify:
(sec^2theta-1)/sec^2theta=sin^2theta
Answers may vary but here is one way:
(sec^2theta-1)/sec^2theta=((tan^2theta+1)-1)/sec^2theta
=tan^2theta/sec^2theta
=tan^2theta(cos^2theta)
=(sin^2theta)/(cos^2theta)(cos^2theta)
=sin^2theta
Solve
2sinx-1=0
x=pi/6+2npi
and
x=(5pi)/6+2npi
where n is an integer
Find the exact value of
sin(u+v)
given
sinu=4/5
,where
0<u<pi/2
and
cosv=-12/13
,where
pi/2<v<pi
-33/65
Solve:
cos2x+cosx=0
x=pi/3+2npi, (5pi)/3+2npi, pi+2npi
where n is an integer
Find the quotient identity:
cot u
(cos u)/(sin u)
Verify:
2sec^2beta=1/(1-sinbeta)+1/(1+sinbeta)
Answers may very but here is one way:
1/(1-sinbeta)+1/(1+sinbeta)=(1+sinbeta+1-sinbeta)/((1-sinbeta)(1+sinbeta))
=2/(1-sin^2beta
=2/cos^2beta
=2sec^2beta
Solve:
cotxcos^2x=2cotx
x=pi/2+npi
where n is an integer
Write
cos(arctan1+arccosx)
as an algebraic expression.
(xsqrt2-sqrt(2-2x^2))/2
Derive a triple-angle formula for
cos3x
cos3x=4cos^3x-3cosx
sin(-u)=?
-sinu
Verify:
(tan^2x+1)(cos^2x-1)=-tan^2x
Answers may vary but here is one way:
(tan^2x+1)(cos^2x-1)=(sec^2x)(-sin^2x)
=-(sin^2x)/(cos^2x)
=-(sinx/cosx)^2
=-tan^2x
Find all the solutions of
2sin^2x-sinx-1=0
in the interval
[0,2pi).
x=(7pi)/6, (11pi)/6, pi/2
Prove the cofunction identity
cos(pi/2-x)=sinx
cos(pi/2-x)=cos(pi/2)cosx+sin(pi/2)sinx
=(0)(cosx)+(1)(sinx)
=sinx
Rewrite
tan^4x
as a quotient of first powers of the cosines of multiple answers.
(3-4cos2x+cos4x)/(3+4cos2x+cos4x)
cos(pi/2-u)=?
sin u
Verify:
tanx+cotx=secxcscx
Answers may vary but here is one way:
tanx+cotx=(sinx)/cosx+cosx/sinx
=(sin^2x+cos^2x)/(cosxsinx)
=1/(cosxsinx)
=1/cosx*1/sinx
=secxcscx
3tan(x/2)+3=0
x=(3pi)/2+2npi
where n is an integer
Simplify:
tan(theta+3pi)
tantheta
Find the exact value of
cos105^o
. Use the half-angle formula to solve.
-(sqrt(2-sqrt3))/2
The pythagorean identity involving sine and cosine.
sin^2u+cos^2u=1
Verify:
csc^4xcotx=csc^2x(cotx+cot^3x)
Answers may vary, but here is one way:
csc^4xcotx=csc^2xcsc^2xcotx
=csc^2x(1+cot^2x)cotx
=csc^2x(cotx+cot^3x)
Solve
sec^2x-2tanx=4
x=arctan3+npi
and
x=-pi/4+npi
where n is an integer
Find all solutions of
sin(x+pi/4)+sin(x-pi/4)=-1
in the interval
[0,2pi)
x=(5pi)/4
and
x=(7pi)/4
Find the exact value of
sin195^o +sin105^o
sqrt2/2