# SOLVING TRIGONOMETRIC EQUATIONS

The equations containing trigonometric functions of unknown angles are known as trigonometric equations. A solution of trigonometric equation is the value of unknown angle that satisfies the equation.

General Solution :

The solution of a trigonometric equation giving all the admissible values obtained with the help of periodicity of a trigonometric function is called the general solution of the equation.

Principal Solution

The smallest numerical value of unknown angle satisfying the equation in the interval [0, 2π] (or) [−π, π] is called a principal solution.

Principal value of sine function lies in the interval

[−π/2, π/2]

Principal value of cosine function is in

[0, π]

Principal value of tangent function is in

(-π/2, π/2)

## Practice Questions

Question 1 :

Solve the following equation for which solutions lies in the interval 0° ≤ θ < 360°

sin4 x  =  sin2 x

Solution :

sin4 x = sin2 x

sin4 x - sin2 x  =   0

sin2 x (sin2 x - 1)  =  0

 sin2 x  =  0 x = sin-1(0)x = 0, π, 2π, ...... sin2 x - 1  =  0sin2 x =  1sin x = √1x = sin-1(1) (or) x = sin-1 (-1)x = π/2 and x = 3π/2

Since we choose the values between 0 to 360, the solution will be {0, π/2, π, 3π/2}.

Question 2 :

Solve the following equation for which solutions lies in the interval 0° ≤ θ < 360°

2 cos2 x + 1  = -3 cos x

Solution :

2 cos2 x + 1  =  -3 cos x

2 cos2 x + 1 + 3 cos x  =  0

Let t = cos x

2t2 + 3t + 1  =  0

By factoring the quadratic equation, we get

(2t + 1)(t + 1)  =  0

 2t + 1  =  02t  =  -1t  =  -1/2cos x  =  -1/2 t + 1  =  0t = -1cos x  = -1

For cos x  =  -1/2

For negative values of cos, we have to select the angle from 2nd and 3rd quadrants.

θ = π - a and π + a

x = π - (π/3)           x = π + (π/3)

x = 2π/3                   x = 4π/3

cos x  =  -1

x = π

So, the solution is {π, 2π/3, 4π/3}.

Question 3 :

Solve the following equation for which solutions lies in the interval 0° ≤ θ < 360°

2 sin2 x + 1  =  3sin x

Solution :

2 sin2 x+1 = 3sinx

Let t = sin x

2 sin2 x - 3sinx + 1 =  0

2t2 - 3t + 1  =  0

By factoring the quadratic equation, we get

(2t - 1)(t - 1)  =  0

 2t - 1  =  02t  =  1t  =  1/2sin x  =  -1/2 t - 1  =  0t  =  1sin x  =  1

For sin x  =  1/2

x = π/6

For positive value of sin, we have to select angles from 2nd quadrant

x = π - a

x = π - (π/6)

x = 5π/6

For sin x  = 1

x  =  π/2

So, the required solution is { π/2,  π/6,  5π/6}

Question 4 :

Solve the following equation for which solutions lies in the interval 0° ≤ θ < 360°

cos 2x  =  1 - 3 sin x

Solution :

cos 2x  =  1 - 3 sin x

1 - 2 sin2x  =  1 - 3 sin x

2sin2x - 3 sin x + 1 - 1  =  0

2sin2x - 3 sin x  =  0

sin x (2 sin x - 3)  =  0

sin x  =  0          2 sin x - 3  = 0

x  =  sin-1(0)          2 sin x  =  3

x  =  0, π          sin x  =  3/2

So, the solution is {0, π}. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 