Show:
Inverse Trig Functions I
Inverse Trig Functions II
Trigonometric Equations I
Trigonometric Equations II
Review
100
Evaluate arcsin(1)
sin = y/1... y=1... Answer: π/4
100
Evaluate sin(2cos(√2/2))
sin(2(π/4)... sin(π/2)... Answer:1
100
Solve from (0,2π) tan x = -1
(-√2/2)/√2/2... Answer: x = 3π/4, 7π/34
100
Solve this equation from (0-2π) 2cosx-1=0
2cosx=1.. cosx=1/2 π/3 , 5π/3
100
what are two ways to write π/4 in radian form
9π/4 -7π/4
200
Evaluate arctan(√3/3)
y/x= (1/2)/(√3/2)= 1/√3= √3/3... y=1/2... x=√3/2... Answer: π/6
200
Evaluate cos(arctan 1 - arcsin1)
cos(π/4-π/2)... (cos π/4)(cos π/2)+(sin π/4)(sin π/2)... (√2/2)(0)+(√2/2)(1)... Answer:√2/2
200
Solve from (0,2π) sec²x = 4
secx=2 or -2... cosx=1/2 or-1/2... Answer: (π/3),(2π/3),(4π/3),(5π/3)
200
Solve this equation cos2x=cos²x
2cos²x-1=cos²x cos²x-1=0 (cosx-1)(cosx+1) x=0 x=π
200
what is tan2x
(2tanx)/(1-tan²x)
300
Evaluate sin(arcsin 1/4)
sin and arcsin cancel each other out therefore... Answer: 1/4
300
Evaluate tan(arccot x/5)
tan and arccot produce a reciprocal therefore... Answer: 5/x
300
Solve from (0,2π) sec²(1/2x) = 4
sec(1/2x)=2 or -2... cos(1/2x)=1/2 or-1/2... 1/2x =(π/3),(2π/3),(4π/3),(5π/3)... Answer: (2π/3),(4π/3)
300
Solve this equation from (0/2π) sinx=cos2x
cos2x= 1-2sin²x=sinx 1-2sin²x-sinx=0 -1(1-2sin²x-sinx) 2sin²x+sinx-1 (2sinx-1)(sinx+1) sinx=1/2 sinx=-1 x= π/6, 5π/6 x= 3π/2
300
what is the sum and difference formula for (tan x+y)
(tanx+tany)/1+tanx tany
400
Evaluate cos(arctan 1)
cos(y/x=1)... cos((√2/2)/(√2/2))... cos(π/4)... Answer: √2/2
400
tan(arcsinx) = ?
Evaluate using a triangle making the side a equal to x and side c equal to 1, therefore making side b equal to √1-x²... Since tan is equal to a/b... Answer: x/√1-x²
400
Solve from (0,2π) sin²x=sinx
sin²x-sinx=0... sinx(sinx-1)=0... sinx = 0 ||| sinx=1... Answer: x= 0,π ||| x= π/2
400
solve 2sin²(x/2)-3sin(x/2)+1=0
(2sin(x/2)-1)(sin(x/2)-1)=o sin(x/2)=1/2 sin(x/2)=1 x/2= π/6, 5π/6 x/2= π/2 x=π/3,5π/3 x=π
400
prove tanx/secx=sinx
(sin/cos)/ (1/cos)... (sin/cos) (cos)... Answer: sin
500
Evaluate sec(arccos -√3/2)
arccos -√3/2 = 5π/6 (x = -√3/2)... sec(5π/6)... sec=1/cos = 1/x... 1/(-√3/2)... 2/-√3... Answer: (-2√3)/3
500
tan(arcsec x/2) = ?
Solve using a triangle making side a equal 2 and side c equal x...After using Pythagorean theorem to get side b we know that the solution is the equivalent of b/a therefore... Answer: √x²-4/2
500
Solve from (0,2π) 4sin²[π/3(x-1)]=3
sin[π/5(x-1)] = √3/2 or -√3/2... π/5(x-1) = (π/3),(2π/3),(4π/3),(5π/3)... x-1=5/3,10/3... Answer: x=8/3,13/3
500
cos²(x/2)=(1/2)sinx
(1+cosx)/2=(1/2)sinx (1+cosx)²=(sinx)² cos²x+2cosx+1=sin²x cos²x+2cosx+1=1/cos²x 2cos²x+2cosx=0 2cosx(cosx+1)=0 cosx=0 cosx=-1 x=π/2, 3π/2 x=π
500
prove 2sin²x-1=sin^4(x)-cos^4(x)
(sin²x+cos²x)(sin²x-cos²x)... 2sin²x-1= (1) sin²x-cos²... 2 (1-cos2x)/2=/1... 1-cos(2x)=-1... -cos(2x)... (-cos²x-sin²x)... sin²x-cos²x=sin²x-cos²x