Review Exercises
Simplifying and Verifying Trigonometric Identities
For the following exercises, find all solutions exactly that exist on the interval [ 0,2π ). [ 0,2π ).
csc 2 t=3 csc 2 t=3
cos 2 x= 1 4 cos 2 x= 1 4
2sinθ=−1 2sinθ=−1
tanxsinx+sin( −x )=0 tanxsinx+sin( −x )=0
9sinω−2=4 sin 2 ω 9sinω−2=4 sin 2 ω
1−2tan(ω)= tan 2 (ω) 1−2tan(ω)= tan 2 (ω)
For the following exercises, use basic identities to simplify the expression.
secxcosx+cosx− 1 secx secxcosx+cosx− 1 secx
sin 3 x+ cos 2 xsinx sin 3 x+ cos 2 xsinx
For the following exercises, determine if the given identities are equivalent.
sin 2 x+ sec 2 x−1= ( 1− cos 2 x )( 1+ cos 2 x ) cos 2 x sin 2 x+ sec 2 x−1= ( 1− cos 2 x )( 1+ cos 2 x ) cos 2 x
tan 3 x csc 2 x cot 2 xcosxsinx=1 tan 3 x csc 2 x cot 2 xcosxsinx=1
Sum and Difference Identities
For the following exercises, find the exact value.
tan( 7π 12 ) tan( 7π 12 )
cos( 25π 12 ) cos( 25π 12 )
sin(70°)cos(25°)−cos(70°)sin(25°) sin(70°)cos(25°)−cos(70°)sin(25°)
cos(83°)cos(23°)+sin(83°)sin(23°) cos(83°)cos(23°)+sin(83°)sin(23°)
For the following exercises, prove the identity.
cos( 4x )−cos( 3x )cosx= sin 2 x−4 cos 2 x sin 2 x cos( 4x )−cos( 3x )cosx= sin 2 x−4 cos 2 x sin 2 x
cos(3x)− cos 3 x=−cosx sin 2 x−sinxsin(2x) cos(3x)− cos 3 x=−cosx sin 2 x−sinxsin(2x)
For the following exercise, simplify the expression.
tan( 1 2 x )+tan( 1 8 x ) 1−tan( 1 8 x )tan( 1 2 x ) tan( 1 2 x )+tan( 1 8 x ) 1−tan( 1 8 x )tan( 1 2 x )
For the following exercises, find the exact value.
cos( sin −1 ( 0 )− cos −1 ( 1 2 ) ) cos( sin −1 ( 0 )− cos −1 ( 1 2 ) )
tan( sin −1 ( 0 )+ sin −1 ( 1 2 ) ) tan( sin −1 ( 0 )+ sin −1 ( 1 2 ) )
Double-Angle, Half-Angle, and Reduction Formulas
For the following exercises, find the exact value.
Find sin( 2θ ),cos( 2θ ), sin( 2θ ),cos( 2θ ), and tan( 2θ ) tan( 2θ ) given cosθ=− 1 3 cosθ=− 1 3 and θ θ is in the interval [ π 2 ,π ]. [ π 2 ,π ].
Find sin( 2θ ),cos( 2θ ), sin( 2θ ),cos( 2θ ), and tan( 2θ ) tan( 2θ ) given secθ=− 5 3 secθ=− 5 3 and θ θ is in the interval [ π 2 ,π ]. [ π 2 ,π ].
sin( 7π 8 ) sin( 7π 8 )
sec( 3π 8 ) sec( 3π 8 )
For the following exercises, use Figure 1 to find the desired quantities.
sin(2β),cos(2β),tan(2β),sin(2α),cos(2α),and tan(2α) sin(2β),cos(2β),tan(2β),sin(2α),cos(2α),and tan(2α)
sin( β 2 ),cos( β 2 ),tan( β 2 ),sin( α 2 ),cos( α 2 ),and tan( α 2 ) sin( β 2 ),cos( β 2 ),tan( β 2 ),sin( α 2 ),cos( α 2 ),and tan( α 2 )
For the following exercises, prove the identity.
2cos( 2x ) sin( 2x ) =cotx−tanx 2cos( 2x ) sin( 2x ) =cotx−tanx
cotxcos(2x)=−sin(2x)+cotx cotxcos(2x)=−sin(2x)+cotx
For the following exercises, rewrite the expression with no powers.
cos 2 x sin 4 (2x) cos 2 x sin 4 (2x)
tan 2 x sin 3 x tan 2 x sin 3 x
Sum-to-Product and Product-to-Sum Formulas
For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.
cos( π 3 )sin( π 4 ) cos( π 3 )sin( π 4 )
2sin( 2π 3 )sin( 5π 6 ) 2sin( 2π 3 )sin( 5π 6 )
2cos( π 5 )cos( π 3 ) 2cos( π 5 )cos( π 3 )
For the following exercises, evaluate the sum by using a product formula. Write the exact answer.
sin( π 12 )−sin( 7π 12 ) sin( π 12 )−sin( 7π 12 )
cos( 5π 12 )+cos( 7π 12 ) cos( 5π 12 )+cos( 7π 12 )
For the following exercises, change the functions from a product to a sum or a sum to a product.
sin(9x)cos(3x) sin(9x)cos(3x)
cos(7x)cos(12x) cos(7x)cos(12x)
sin(11x)+sin(2x) sin(11x)+sin(2x)
cos(6x)+cos(5x) cos(6x)+cos(5x)
Solving Trigonometric Equations
For the following exercises, find all exact solutions on the interval [ 0,2π ). [ 0,2π ).
tanx+1=0 tanx+1=0
2sin(2x)+ 2 =0 2sin(2x)+ 2 =0
For the following exercises, find all exact solutions on the interval [ 0,2π ). [ 0,2π ).
2 sin 2 x−sinx=0 2 sin 2 x−sinx=0
cos 2 x−cosx−1=0 cos 2 x−cosx−1=0
2 sin 2 x+5sinx+3=0 2 sin 2 x+5sinx+3=0
cosx−5sin( 2x )=0 cosx−5sin( 2x )=0
1 sec 2 x +2+ sin 2 x+4 cos 2 x=0 1 sec 2 x +2+ sin 2 x+4 cos 2 x=0
For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval [0,2π). [0,2π). Round to four decimal places.
3 cot 2 x+cotx=1 3 cot 2 x+cotx=1
csc 2 x−3cscx−4=0 csc 2 x−3cscx−4=0
For the following exercises, graph each side of the equation to find the approximate solutions on the interval [0,2π). [0,2π).
20 cos 2 x+21cosx+1=0 20 cos 2 x+21cosx+1=0
ਸੈਕ² x − 2ਸੈਕ x = 15