Trigonometry

Question 1

Evaluate the limit

lim_x~>0[sin3x/sin4x]

Answer 1

3/4

Times both denumerator and denumerator with 4x/4x and 3x/3x

Question 2

Evaluate the limit

lim_x~>0[(1-cosx)/x]

Answer 2

0

Question 3

Evaluate the limits

lim_x~>π[sinx/sin(π-x)]

Answer 3

1

sin(pi-x) = sin x

Question 4

Derive

sin(x)

Answer 4

cos(x)

Question 5

Derive

cos(x)

Answer 5

-sin(x)

Question 6

Derive using first principles

sin(x)

Answer 6

cos(x)

Employs additive angles indentities.

Question 7

Find the tangent line equation in the form Ax+By+C=0, at x = π/6

y = sin(x)/cos(2x)

Answer 7

0 = 6x√3+2y+2-π√3

Question 8

Derive

y = 2cos(3πx)

Answer 8

-6πsin(3πx)

Question 9

Derive

y=sin⁴(x)

Answer 9

4sin³(x)cos(x)

Question 10

Derive

y=(cos3x)/(1-cos3x)

Answer 10

dy/dx=-3sin3x/(1-cos3x)²

Question 11

Find dy/dx for

x = sin y

Answer 11

sec y

Use implicit differentiation, such that sin y becomes cos y dy/dx

Question 12

Derive

tan x

Answer 12

sec²x

Question 13

Derive

sec x

Answer 13

sec x tan x

Question 14

Derive

cot x

Answer 14

-csc²x

Question 15

Derive

csc x

Answer 15

-csc x cot x

Question 16

Find dy/dx

tan y =x²

Answer 16

2xcos²y

Question 17

Find the tangent radiant at x for

y = tan( csc(x) )
in the interval (0, π/2)
knowing sin x = 1/π

Answer 17

-π√(π²-1)

Substitute sin x = 1/π into the derivatives

Question 18

Prove that the included function concave upward

y = sec x + tan x

for the interval (-π/2, π/2)

Answer 18

After deriving the equation twice, take every sec³ x out and factor it. The second derivatives should be sec³ x (sin x +1)²

Question 19

Let x be the angle between two sides (each 50 cm) of a 5 m long trough. Find x where the trough’s volume is maximized.

Answer 19

x = π/2

Question 20

Triangle ABC is imposed into a semi-circle with diameter AB of 10 cm. Find the angle at A that would maximized the area.

Hint: use the semi circle equations.

Answer 20

π/4

Question 21

sin(π/6)

Answer 21

1/2