Chapter 5 Analytic Trigonometry

Fundamental Identities

100

Find the reciprocal identity:

csc u

1/sin u

200

Find the quotient identity:

cot u

(cos u)/(sin u)

300

sin(-u)=?

-sinu

400

cos(pi/2-u)=?

sin u

500

The pythagorean identity involving sine and cosine.

sin^2u+cos^2u=1

Verifying Trig Identities

100

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

200

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

300

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

400

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

500

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)

Solving Trig Equations

100

Solve 

2sinx-1=0

x=pi/6+2npi

and

x=(5pi)/6+2npi

where n is an integer

200

Solve:

cotxcos^2x=2cotx

x=pi/2+npi

where n is an integer

300

Find all the solutions of

2sin^2x-sinx-1=0

 in the interval 

[0,2pi).

x=(7pi)/6, (11pi)/6, pi/2

400

3tan(x/2)+3=0

x=(3pi)/2+2npi

where n is an integer

500

Solve 

sec^2x-2tanx=4

x=arctan3+npi

and

x=-pi/4+npi

where n is an integer

Sum & Difference Formulas

100

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

200

Write

cos(arctan1+arccosx)

 as an algebraic expression.

(xsqrt2-sqrt(2-2x^2))/2

300

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

400

Simplify:

tan(theta+3pi)

tantheta

500

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

Multiple-Anlge & Product-to-Sum

100

Solve:

cos2x+cosx=0

x=pi/3+2npi, (5pi)/3+2npi, pi+2npi

where n is an integer

200

Derive a triple-angle formula for 

cos3x

cos3x=4cos^3x-3cosx

300

Rewrite

tan^4x

 as a quotient of first powers of the cosines of multiple answers.

(3-4cos2x+cos4x)/(3+4cos2x+cos4x)

400

Find the exact value of

cos105^o

. Use the half-angle formula to solve.

-(sqrt(2-sqrt3))/2

500

Find the exact value of 

sin195^o +sin105^o

sqrt2/2