mathacle pset algebra trig functions level 2...

Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1

Name:___________ Date: ___________ I. TRIG IDENTITIES Some of the useful identities

sintan

cos

xx

x=

2 2sin cos 1x x+ =

� Prove each identity 1.) 2(cos )(tan sin cot ) sin cosx x x x x x+ = +

2.) ( )2 21 tan sec 2 tanx x x− = −

3.) 22

(1 cos )(1 cos )tan

cos

u uu

u

− + =

4.) 2cos 1

tan sincos

xx x

x

− = −

5.) ( ) ( )2 2cos sin cos sin 2t t t t− + + =

6.) 2

22 2

1 tansec

sin cos

xx

x x

+ =+


Name:___________ Date: ___________

7.) cos 1 sin

1 sin cos

x x

x x

−=+

8.) 2tan 1 cos

sec 1 cos

x x

x x

−=+

9.) 2 2 2 2cot cos cos cotx x x x− = 10.) 4 4 2 2cos sin cos sinx x x x− = −

11.) sin 1

tan seccos

xx x

x

− = −

12.) 2

tan sin tan

sin 1 cos

x x x

x x

− =+

13.) sec csc

sin costan cot

x xx x

x x

+ = ++


Name:___________ Date: ___________ 14.) 4 4 2sec tan 1 2 tanx x x− = +

15.) 1 cos

tansin cos sin

xx

x x x− =

16.) sec sin

cotsin cos

x xx

x x− =

17.) 22

cos coscot

sec 1 tan

x xx

x x− =

−

18.) 3 2 3 5sin cos sin sinx x x x= − 19.) (1 sin cos )(1 sin cos ) 2sin cosx x x x x x+ + − − = − 20.) ( ) ( )6 4 5 3sec sec tan sec sec tan sec tanx x x x x x x x− =


Name:___________ Date: ___________

21.) 21 12sec

1 sin 1 sinx

x x+ =

+ −

22.) 1 1

2 tantan sec tan sec

xx x x x

+ = −− +

23.) sin cos

sin coscos sin

1 1sin cos

x xx x

x x

x x

+ = +− −

24.) xxx

x

x

xcsccot4

cos1

cos1

cos1

cos1 =+−−

−+

25.) 4 4

2 2

sin cos1

sin cos

x x

x x

− =−

26.) cos 1 sin

1 sin cos

x x

x x

+=−


Name:___________ Date: ___________

27.) sec tan 1 sin

secx tan 1 sin

x x x

x x

+ +=− −

28.) 3 3 2 2

2

sin cos csc cot 2cos

sin cos 1 cot

x x x x x

x x x

− − −=+ −

29.) 1csc2cos1

cos

cos1

1 2 +=+

−−

xx

x

x

30.) ( ) ( )( )1cos1sin21cossin 2 ++=++ xxxx


Name:___________ Date: ___________ II. SUM, DIFFERENCE AND MULTIPLE-ANGLE IDENTITIES Useful identities: sin( ) sin cosy sin cosx y x y x± = ± , cos( ) cos cosy sin sinx y x x y± = ∓ ,

tan tantan( )

1 tan tan

x yx y

x y

±± =∓

, 1 cos

sin2 2

x x−= ± , 1 cos

cos2 2

x x+= ±

� Find exact value for each trig expression

1.) sin15o 2.) cos 75o

3.) 7

cos12

π

4.) 11

tan12

π

5.) sin12

π−


Name:___________ Date: ___________ 6.) tan195o 7.) sin 22.5o

8.) cos8

π

9.) 5

sin12

π

10.) 7

tan12

π

� Write the expression as the sine, cosine, or tangent of an angle. No Calculator. 11.) sin 62 cos17 cos 62 sin17o o o o− 12.) cos138 cos18 sin138 sin18o o o o+

13.) sin cos cos sin2 6 2 6

π π π π−


Name:___________ Date: ___________

14.) 5 5 5 5

cos cos sin sin3 6 3 6

π π π π+

15.) tan19 tan 26

1 tan19 tan 26

o o

o o

+−

16.) tan tan

5 20

1 tan tan5 20

π π

π π

+

−

17.) sin(3x)cos(x) cos(3x)sin(x)− 18.) cos(7 y)cos(3y) sin(7 y)sin(3 y)−

19.) tan 3 tan 2

1 tan 3 tan 2

α βα β−

+


Name:___________ Date: ___________

� Prove the identity

21.) ( )sin cos2

x xπ − = −

22.) yxyxyx cossin2)sin()sin( =++− 23.) 3 2cos3 cos 3sin cosx x x x= − 24.) cos3 cos 2cos 2 cosx x x x+ = 25.) sin 4 sin 2 2sin 3 cosx x x x+ =

26.) 2 2

2 2

tan tantan( ) tan( )

1 tan tan

x yx y x y

x y

−+ − =−


Name:___________ Date: ___________

27.) 2 2

2 2

tan 4 tantan 5 tan 3

1 tan 4 tan

u uu u

u u

−=−

28.) sin( ) tan tan

sin( ) tan tan

x y x y

x y x y

+ +=− −

29.) cos( ) cos cos( ) 1 sin( )

cos sinx h x h h

x xh h h

+ − − = + ⋅

30.) 4 4cos sin cos 2x x x− =

31.) ( )3sin sin sin cos

2x x x x

π π + + − = −


Name:___________ Date: ___________

� Express the function as a sinusoid in the form of sin( )y A Bx C= + , or tan( )y A Bx C= + . No Calculator.

32.) 3 sin cosy x x= +

33.) 1 tan

1 tan

xy

x

+=−

� Solve the indicated variable

34.) Given 3

cos5

x = and sin 0x < , find sin 2x , cos 2x and tan 2x .

35.) Given 5

sin13

x = and x is in Quadrant II, find sin 2x , cos 2x and tan 2x .

36.) Given cot 4x = − and x is in Quadrant IV, find sin 2x , cos 2x and tan 2x .


Name:___________ Date: ___________ 37.) Solve the equation sin 2 cosx x= , where [0, 2 )x π∈ 38.) Solve the equation cos 2 cosx x= , where [0 ,360 )o ox ∈ 39.) Solve the equation sin 2 tan 0x x− = , where [0, 2 )x π∈


Name:___________ Date: ___________ III. TRIG EQUATIONS

� Find x in the interval[0, 2 )π . 1.) 2sin 1x =

2.) 2sec 2 2 0x − =

3.) 2 3cos

4x =

4.) 3cos

cotsin

xx

x=

5.) 22sin sin 1x x+ =


Name:___________ Date: ___________ 6.) 2cos sin cos 0x x x− = 7.) 2 22 tan sin tan 0x x x− = 8.) 3 22cos cos cos 0x x x+ − =

� Find x in the interval 0[0 , 360 )o . 9.) 2cos 0x = 10.) 2sin 0.25 0x − = 11.) 24sin 3 0x − =


Name:___________ Date: ___________

15

12.) 2sin cos sin 0x x x+ = 13.) 22cos 3cos 2 0x x+ − = 14.) sec 2cos 1 0x x− + = 15.) sin tan sinx x x=

16.) 3 tan 3 0x − = 17.) 4 23cos 4cos 0x x+ =


Name:___________ Date: ___________

16

18.) 22sin 1 sinx x− = 19.) 2cos 2sec 3 0x x− − =

20.) sin 3 cos 0x x− = 21.) 23sin 5sin 2 0x x+ − =

22.) 2

cos 22

x =


Name:___________ Date: ___________

17

23.) sin 2 x 0=

24.) 3

sin 3x2

=

25.) 3sin 2 sin 2 0x x− =

26.) 2

1)2cos(

−=x

27.) 04tan =x 28.) 0cos2cos =+ xx

29.) 32

sin32 =x

30.) xx 2cos1cos =−

31.) 2

3sin2sincos2cos

−=− xxxx


Name:___________ Date: ___________

18

Answers

1.) 3

,2 2

π π

2.) 5 3

, ,6 6 2

π π π

3.)

4.) 5

,6 6

π π

5.) 6.) 7.) 0,180 , 45 , 225o o o

8.) 150 , 330o o


Name:___________ Date: ___________

19

IV. INVERSE TRIG FUNCTIONS

Always true for ( ) ( ) ( )1 1 1sin sin ( ) , cos cos ( ) , tan tan ( )x x x x x x− − −= = =


Name:___________ Date: ___________

20

� Find the exact value of each expression without a calculator

1.) 1 1sin

2− =

______________________________________

2.) 1 3sin

2−

− =

______________________________________

3.) 1sin2

π− =

______________________________________

4.) 1sin sin9

π− =

______________________________________

5.) 1 5sin sin

6

π− =

______________________________________

6.) 1 2cos

2−

− =

______________________________________

7.) 1tan 3− = ______________________________________ 8.) ( )( )1cos cos 1.1− − = ______________________________________


Name:___________ Date: ___________

21

9.) 1tan cos ( )3

π−

______________________________________

10.) 1 3sin tan

4−

______________________________________

11.) 4

sec arcsin5

______________________________________

12.) ( )cos arctan 2 ______________________________________

13.) 1 5sin cos

5−

______________________________________

14.) 5

cos arcsin13

______________________________________

15.) 1 5csc tan

12− −

______________________________________

16.) 3

sec arctan5

−

______________________________________

17.) 5

tan arcsin6

−

______________________________________


Name:___________ Date: ___________

22

� Write an algebraic expression which is equivalent to the given expression

18.) ( )1cot tan x− = ______________________________________

19.) ( )1sin tan x− = ______________________________________

20.) ( )cos arcsin 2x = ______________________________________

21.) ( )1sec tan 3x− = ______________________________________

22.) ( )1sin cos x− = ______________________________________

23.) 1 1cot tan

x− =

______________________________________

24.) tan arccos3

x =

______________________________________

25.) ( )( )1sec sin 1x− − = ______________________________________

26.) 1csc tan2

x− =

______________________________________

27.) x

sin arctan( )2

______________________________________


Name:___________ Date: ___________

23

Answers

1.) 6

π

2.) 3

π−

3.) No soln.

4.) 9

π

5.) 6

π

6.) 3

4

π

7.) 3

π

8.) 1.1

9.) 29 x

x

−

10.) 3

5

11.) 5

3

12.) 5

5

13.) 2 5

5

14.) 12

13

15.) 13

5

−

16.) 34

5

17.) 5 11

11

−

18.) 1

x

19.) 2 1

x

x +

20.) 21 4x−

21.) 21 9x+

22.) 21 x− 23.) x

24.) 29 x

x

−

25.) 2

1

2x x−

26.) 2 2x

x

+

27.) 2 4

x

x +


Name:___________ Date: ___________

24

V. LAW OF SINES In any ABC∆ with angles A, B, and C opposite sides a, b, and c, respectively, the following equation is true:

sin sin sinA B C

a b c= =

Two triangles are congruent if AAS, SAS, ASA or SSS. Two triangles are similar if AA. The ambiguous case is ASS, where zero, one or two triangles could satisfy ASS.

� Solve each triangle 1.) 040A = , 030B = , 10b = ______________________________________ 2.) 033A = , 070B = , 7b = ______________________________________


Name:___________ Date: ___________

25

3.) 032A = , 17a = , 11b = ______________________________________ 4.) 045B = , 28a = , 27b = ______________________________________ 5.) 070B = , 9b = , 14c = ______________________________________ 6.) 0103C = , 46b = , 61c = ______________________________________


Name:___________ Date: ___________

26

7.) Find m A∡ to the nearest whole degree. ___________________________

8.) Find m DGF∡ to the nearest whole degree. ___________________________

9.) Find BC to the nearest whole number. 15CD =


Name:___________ Date: ___________

27

VI. LAW OF COSINES In any ABC∆ with angles A, B, and C opposite sides a, b, and c, respectively, the following equation is true:

2 2 2

2 2 2

2 2 2

2 cos

2 cosB

2 cosC

a b c bc A

b a c ac

c a b ab

= + −= + −= + −

� Solve for each triangle

1.) 041.4A = , 2.78b = , 3.92c = ______________________________________ 2.) 074.80B = , 8.919a = , 6.427c = ______________________________________ 3.) 1240AB = , 876AC = , 965BC = ______________________________________ 4.) 4a = , 5b = , 8c = ______________________________________


Name:___________ Date: ___________

28

VII. AREA OF A TRIANGLE The area of a triangle is

1 1 1sin sin sin

2 2 2A ab C ac B bc A= = =

( )( )( )A s s a s b s c= − − −

where 2

a b cs

+ += .

� Solve area for each triangle

1.) 084C = , 32a = , 37c = ______________________________________ 2.) 029A = , 49b = , 50c = ______________________________________ 3.) 11a = , 14b = , 20c = ______________________________________ 4.) 5a = , 7b = , 10c = ______________________________________ 5.) 8a = , 12b = , 28c = ______________________________________


Name:___________ Date: ___________

29

6.) ______________________________________

7.) ______________________________________


Name:___________ Date: ___________

30

VIII. ANGLES BETWEEN INTERSECTING LINES The inclination of a line is measured by the positive angle formed by the x-axis to the line. The angle is always between zero and 180 degrees.

For line 1L : 1 1y m x b= + , 1 1tanm α= . For line 2L : 2 2y m x b= + , 2 2tanm α= . Use the

larger of 1 2,m m to subtract the smaller of 1 2,m m to calculate the acute intersecting angle.

2 1 2 12 1

2 1 2 1

tan tantan( ) tan( )

1 tan tan 1

m m

m m

α αθ α αα α− −= − = =

+ +

Or

1 2 1

2 1

tan1

m m

m mθ − −= +

, 180oφ θ= −

Where 2 1 1m m ≠ − and 2m and 1m are finite. When 2 1 1m m = − , the angle is 90o . When

either 2m or 1m is infinite, the angle is complement of either 2α or 1α .

� Find the tangent of the angle between the lines whose slopes are given below.

1.) 1

2and

2

3

2.) 3

4

−and

5

2

−


Name:___________ Date: ___________

31

3.) 2

7

−and

5

3

4.) m and 0

� To the nearest minute, find the angle between the lines whose slopes are given below

5.) 5

2and

2

3

6.) 1.3− and 0.6

7.) m = ∞ (no slope) and 1

2−

8.) The tangent of the angle between two lines is 4

9− , and the slope of the line with the

smaller inclination is 3

7. Find the slope of the other line.

9.) To the nearest degree decimal, find the interior angles of the triangle whose vertices are (-2,-3), (-5,4) and (6,1).


Name:___________ Date: ___________

32

10.) Show that the triangle whose vertices are (-2,3), (6,9), and (4,11) is isosceles. 11.) Denote the line through (2,1) and (4,3) by q. What is the slope of a line p such that the angle between q and p is 45 degrees?

mathacle pset algebra trig functions level 2...

Documents