Question

5. Find the complete exact solution of sin x = -√3/2.

9. Solve cos 2x – 3sin x cos 2x = 0 for the principal value(s) to two decimal places.

21. Solve tan^2x + tan x – 1 = 0 for the principal value(s) to two decimal places.

22. Prove that tan^2a – 1 + cos^2a = tan^2a sin^2a


Answer 1

5. From the 30-60-90° triangle we know
sin 60° = +√3/2
so x must be in quadrants III or IV
x = 180+60= 240° or x = 360-60 = 300°
or x = 4π/3 or x = 5π/3

9.
cos 2x(1 - 3sinx) = 0
cos 2x = 0 or sinx = 1/3

for cos 2x = 0
2x = 90° or 2x = 270°
x = 45° or x = 135°
since the period of cos 2x is 180° , two other answers are 225° and 315°
x = π/4 , 3π/4 , 5π/4 and 7π/4

for sinx = 1/3
x = 19.47° or x = 160.53°

10. Using the quadratic equation we have
tan x = (-1 ±√5)/2
if tanx = (-1+√5)/2 , then x = 31.72° or 211.72°
if tanx = (-1 - √5)/2 , then x = 121.72 or 301.72°

22.
LS = tan^2 a - 1 + cos^2 a
= sec^2 a -1 -1 + cos^2 a
= 1/cos^2 a - 2 + cos^2 a
= (1 - 2cos^2 a + cos^4 a)/cos^2 a
= (cos^2 a - 1)^2 /cos^2 A
= (-sin^2 a)^2 / cos^2 a
= sin^4 a / cos^2 a
= sin^2 a / cos^2 a (sin^2 a)
= tan^2 a sin^2 a
= RS