2 mathematics Q

1) cos2 x + cos x - sin x = 0

cos2 x + cos x = sin x

cos4 x + 2 cos3 x + cos2 x = sin2 x = 1 - cos2 x

cos4 x + 2 cos3 x + 2 cos2 x - 1 = 0

cos x = -1 or 0.544

x = -π, π, 0.996 or -5.287

2) y = sin x/(1 + tan x)

Note that when tan x tends to -1, y can be very large (+∞) or very small (-∞)

In quadrant 2: When x < 3π/4, tan x < -1 and hence 1 + tan x < 0.

So when x tends to 3π/4 from x < 3π/4, y will tend to -∞ since sin x > 0

Then when x tends to 3π/4 from x > 3π/4, y will tend to +∞.

Similar approach for x tends to -5π/4 which is also in quadrant 2.

In quadrant 4: When x < 7π/4, tan x < -1 and hence 1 + tan x < 0.

So when x tends to 7π/4 from x < 7π/4, y will tend to +∞ since sin x < 0

Then when x tends to 7π/4 from x > 7π/4, y will tend to -∞.

Similar approach for x tends to -π/4 which is also in quadrant 4.

1) cos2 x + cos x - sin x = 0

cos2 x + cos x = sin x

cos4 x + 2 cos3 x + cos2 x = sin2 x = 1 - cos2 x

cos4 x + 2 cos3 x + 2 cos2 x - 1 = 0

cos x = -1 or 0.544

x = -π, π, 0.996 or -5.287

2) y = sin x/(1 + tan x)

Note that when tan x tends to -1, y can be very large (+∞) or very small (-∞)

In quadrant 2: When x < 3π/4, tan x < -1 and hence 1 + tan x < 0.

So when x tends to 3π/4 from x < 3π/4, y will tend to -∞ since sin x > 0

Then when x tends to 3π/4 from x > 3π/4, y will tend to +∞.

Similar approach for x tends to -5π/4 which is also in quadrant 2.

In quadrant 4: When x < 7π/4, tan x < -1 and hence 1 + tan x < 0.

So when x tends to 7π/4 from x < 7π/4, y will tend to +∞ since sin x < 0

Then when x tends to 7π/4 from x > 7π/4, y will tend to -∞.

Similar approach for x tends to -π/4 which is also in quadrant 4.