mathacle pset algebra trig functions level 2...
TRANSCRIPT
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
+ =+
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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
+ = ++
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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− =
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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
+=−
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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−
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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−
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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−
+
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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
−+ − =−
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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 + + − = −
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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 .
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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∈
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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+ =
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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 − =
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 3
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+ =
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 3
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 =
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 3
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
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 3
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
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 3
Name:___________ Date: ___________
19
IV. INVERSE TRIG FUNCTIONS
Always true for ( ) ( ) ( )1 1 1sin sin ( ) , cos cos ( ) , tan tan ( )x x x x x x− − −= = =
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 3
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− − = ______________________________________
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 3
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
−
______________________________________
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 3
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
______________________________________
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 3
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 +
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 3
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 = ______________________________________
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
Name:___________ Date: ___________
25
3.) 032A = , 17a = , 11b = ______________________________________ 4.) 045B = , 28a = , 27b = ______________________________________ 5.) 070B = , 9b = , 14c = ______________________________________ 6.) 0103C = , 46b = , 61c = ______________________________________
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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 =
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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 = ______________________________________
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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 = ______________________________________
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
Name:___________ Date: ___________
29
6.) ______________________________________
7.) ______________________________________
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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
−
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 1
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).
Mathacle PSet ---- Algebra, Trigonometry Functions Level --- 2 Number --- 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?