wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Trigonometric EquationPrincipal Solutions1. sin x=32A. 5π62. tan x=13B. 5π123. sec 2x=23C. 7π124. cos 3x=(12)D. π3E. 2π9F. 4π9G. 11π6H. 2π3


A

1 - D, 2 - A, 3 - B, 4 - E

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B

1 - D, 2 - G, 3 - B, 4 - F

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C

1 - H, 2 - A, 3 - C, 4 - F

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D

1 - H, 2 - G, 3 - C, 4 - E

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
Open in App
Solution

The correct options are
A

1 - D, 2 - A, 3 - B, 4 - E


B

1 - D, 2 - G, 3 - B, 4 - F


C

1 - H, 2 - A, 3 - C, 4 - F


D

1 - H, 2 - G, 3 - C, 4 - E


We have learnt that the values of sinx & cosx repeat after an interval of 2π and the values of tanx repeat after an interval of pi. The solutions of trigonometric equations for which x lies between 0 and 2π are called principal solutions.

So,

1. sinx=32

We know sinx is positive in 1st & 2nd quadrant in the interval of 0 to 2π.

in 1st quadrant, sinπ3=(3)2

in 2nd quadrant sin(ππ3)=sin2π3=32

Therefore, principal solution of equation sinx=32 are π3,2π3

2. tanx=13

tanx is negative in 2nd & 4th quadrant in the interval of 0 to 2π.

We know that tanπ6=13. tan(ππ6)=tanπ6=13

So, ππ6=5π6 and

in 4th quadrant (2ππ6)=tanπ6=13
(2ππ6)=11π6

Therefore, Principal solutions are 5π6 and 11π6

3. sec 2x=23
1cos2x=23
cos2x=32

cos θ is negative in 2nd & 3rd quadrant in the interval of 0 to 2π.

We know, cosπ6 = 32

in 2nd quadrant cos(ππ6)=cos(π6)=32

Also, ππ6=5π6
2x=5π6
or x=5π12

In 3rd quadrant cos(π+π6)=cosπ6=32
2x=π+π6=7π6
or x=7π12

Therefore, principal solutions of equation sec 2x=23 are 5π12 and 7π12
4. cos3x=12

cosθ is negative in 2nd & 3rd quadrant

cosπ3=12

In 2nd quadrant cos(ππ3)=cosπ3=12
3x=ππ3=2π3
x=2π9

In 3rd quadrant cos(π+π3)=cosπ3=12
3x=π+π3=4π3
x=4π9

Therefore, principal solutions of equation cos 3x=12 are 2π9 and 4π9.


flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Principal Solution
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon