Unit Circle
Pythagorean Identities
Proving Identities
miscellaneous
100
sin(π/3)
√3
Explanation- Unit Circle
100
If sin(x)= 3/8 and x is in Quadrant I, Find cos(x) exactly
√55/8
Explanation-
Cos^2x+(3/8)^2=1
Cos^2x+(9/64)=1
Cos^2x=1-(9/64)
√Cos^2x= (√55/√64)
Cos(x)= √55/8
100
cot2x+1=csc2x
(cos2x/sin2x)+1 I 1/sin2x
(cos2x/sin2x)+(sin2x/sin2x) I
(cos2x/sin2x)/sin2x I
1/sin2x I
100
Give the period of each function
Cos=2π
Sin=2π
Tan=π
Cot=π
Sec=2π
Csc=2π
200
cos(2π/3)
-1/2
explanation- Unit circle
200
If cos x= 2/7, find sin x exactly
√45/7
Explanation-
sin^2x+(2/7)=1
sin^2x+(4/49)=1
sin^2x=1-(4/49)
√sin^2x=√45/√49)
sin(x)=(√45/7)
200
tan(x)*cot(x)=cos2x+sin2x
(sinx/cosx)*(cosx/sinx) I cos2x+sin2x
=1 I =1
I
200
Not Drawn To Scale
a=2cm
b=9cm
B=?
77.47
Explanation-
tan x=9/2
x=tan-1(9/2)
=77.47
300
csc(π/2)
1
Explanation- csc(x)= 1/sin
the sin of π/2= 1. The reciprocal of 1 is 1 causing the csc=1
300
If cos x=5/9, find exactly csc(x)
9/√56
Explanation-
sin^2x+(5/9)=1
sin^2x+(25/81)=1
sin^2x=1-(25/81)
√sin^2x=√56/√81)
sin(x)=(√56/9)
csc(x)= (9/√56)
300
cos x(sec x-cos x)=sin2x
cos x(sec x-cos x) I sin2x
cos x(1/cos x-cos x) I
(1-cos2x) I
sin2x I
300
Find sin(13π/12) exactly.
(√2-√6)/4
Explanation-
Sin(13π/12)= Sin(5π/6+π/4)
sin(5π/6)cos(π/4)+cos(5π/6)sin(π/4)
1/2*√2/2+-√3/2*√2/2
(√2/4)+-(√6/4)
=(√2-√6)/4
400
tan 3π/4
1
Explanation- tan=sin/cos
the sin and cos of 3π/4= 1/2
1/2÷1/2=1
400
If x=5π/6 Find the EXACT VALUES for all six trigonometric functions
cosx=-√3/2 sec=-2/√3
sinx=1/2 csc=2
tan=-√3/3 cot =3/√3
Explanation- Find Cos, Sin, and Tan. Then find their reciprocals
400
cos x/sec x*sin x= csc x-sin x
cos x/sec x*sin x I csc x-sin x
cos x/(1/cos x)*sin x I 1/sin-sin x
cos x/(sin x/cos x) I cos2x/sin x
cos2x/sin x I cos2x/sin x
400
Prove
2sin x+csc (2x)=sec x
2sin x*1/sin 2x I 1/cos x
2/1*sin x/1*1/2sincos I
1/cos x I
500
Cot(19π/6)
√3
Explanation- cot= 1/tan, cos/sin
500
If sin x=3/13 and x is in Quadrant I, find cos x, tan x, sec x, csc x, and cot x.
sin(x)=3/13 csc(x)= 13/3
cos(x)=√160/13 sec(x)=13/√160
tan(x)=3/√160 cot(x)=√160/3
Explanation
Cos^2x+(3/13)^2=1
Cos^2x+(9/169)=1
Cos^2x=1-(9/169)
√Cos^2x= (√160/√169)
Cos(x)= √160/13
Tan= sin/cos, 3/13÷√160/13= 3/√160
Then find reciprocals.
500
cos x*cot x/1-sin x=1+csc x
cos x*cot x/1-sin x I 1+csc x
(cos x*(cos x/sin x))/1-sin x I 1+1/sin x
(cos2x/sin x)/1-sin x I (sin x/sin x)+(1/sinx)
1-sin2x/sin x/(1-sin x) I (sin x +1)/sin x
(1+sin x)/sin x I
500
cos x=9/13 find sin 2x
(36√22)/169
Explanation-
sin2x+(9/13)=1
sin2x+(81/169)=1
sin2=1-(81/169)
√sin2x=√88/√169)
sin(x)=(√88/13)
2*(√88/13)*(9/13)
(36√22)/169