PreCalculus Honors

Unit Test #7

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No Calculator Allowed

 

1.

Use the fundamental identities to simplify cos 2 x tan 2 x+1 as much as possible.

(4)

cos 2 x tan 2 x+1 = cos 2 x sin 2 x cos 2 + cos 2 x cos 2 x = sin 2 x+ cos 2 x=1

 

2.

Verify: 2 sin 2 x cos x =sec x +cos x

(6)

Working the right-hand side…

sec x +cos x = 1 cos x + cos 2 x cos x = 1+ cos 2 x cos x = 1+ 1 sin 2 x cos x = 2 sin 2 x cos x

QED.

 

3.

Verify: cos 2 x 1sin x =1+sin x

(6)

Working the left-hand side…

cos 2 x 1sin x = 1 sin 2 x 1sin x = 1+sin x 1sin x 1sin x =1+sin x

QED.

 

4.

Find the exact value of cos 2π 9 cos π 36 sin 2π 9 sin π 36 .

(4)

This smells like the sum formula for cosine…

cos 2π 9 + π 36 =cos 8π 36 + π 36 =cos 9π 36 =cos π 4 = 1 2

 

5.

Find the exact value of sin 5π 8 .

(5)

This will require the half-angle formula for sine. Note that 5π 8 is in the second quadrant, so the value must be positive.

1cos 5π 4 2 = 1 1 2 2 = 1+ 2 2 2 = 2+ 2 4

 

6.

Solve sin 2x = 1 2 for x 0,2π .

(6)

Note that x 0,2π 2x 0,4π . sin 2x = 1 2 2x 7π 6 , 11π 6 , 19π 6 , 23π 6 x 7π 12 , 11π 12 , 19π 12 , 23π 12

 

7.

Solve sin 2 x +2sin x +1=0 for x 0,2π .

(5)

sin 2 x +2sin x +1= sin x +1 2 ; thus, sin x +1 2 =0 sin x +1 =0sin x =1x= 3π 2 .

 


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