Question

Use a calculator to solve the equation on the interval [0, 2π). Round to the nearest hundredth of a radian. sin 2x - sin x = 0 : 0, 1.05, 3.14, 5.24 1.05, 3.14, 5.24 0, 2.09, 4.19 0, 2.09, 3.14, 4.19

Answer 1

Sin 2x - sin x=0
Using the trigonometric identity: sin 2x=2 sinx cosx
2 sinx cosx - sinx =0
Common factor sinx
sinx ( 2 cosx -1)=0
Two options:
1) sinx=0
on the interval [0,2π), the sinx=0 for x=0 and x=π=3.1416→x=3.14

2) 2 cosx - 1=0
Solving for cosx
2 cosx-1+1=0+1
2 cosx = 1
Dividing by 2 both sides of the equation:
(2 cosx)/2=1/2
cosx=1/2

cosx is positive in first and fourth quadrant:
First quadrant cosx=1/2→x=cos^(-1) (1/2)→x=π/3=3.1416/3→x=1.05
Fourth quadrant: x=2π-π/3=(6π-π)/3→x=5π/3=5(3.1416)/3→x=5.24

Answer: Solutions: x=0, 1.05, 3.14, and 5.24

Answer 2

Solution of the equation sin 2x - sin x = 0 on the interval [0 , 2π) are :

x = { 0 , 1.05 , 3.14 , 5.24 }

Further explanation

Firstly , let us learn about trigonometry in mathematics.

Suppose the ΔABC is a right triangle and ∠A is 90°.

sin ∠A = opposite / hypotenuse

cos ∠A = adjacent / hypotenuse

tan ∠A = opposite / adjacent

There are several trigonometric identities that need to be recalled, i.e.

$cosec ~ A = \frac{1}{sin ~ A}$

$sec ~ A = \frac{1}{cos ~ A}$

$cot ~ A = \frac{1}{tan ~ A}$

$tan ~ A = \frac{sin ~ A}{cos ~ A}$

Let us now tackle the problem!

$\sin 2x - \sin x = 0$

$2 \sin x \cos x - \sin x = 0$

$\sin x (2 \cos x - 1) = 0$

$\sin x = 0 ~ or ~ 2 \cos x - 1 = 0$

If sin x = 0 , then for the interval [0 , 2π) → x = { 0 , 3.14 }

For 2 cos x - 1 = 0  :

2 cos x = 0 + 1

2 cos x = 1

cos x = ½

If cos x = ½ , then for the interval [0 , 2π) → x = { 1.05 , 5.24 }

If we draw a graph from the function above, it will look like the picture in the attachment.

Conclusion :

Solution of the equation sin 2x - sin x = 0 on the interval [0 , 2π) are :

x = { 0 , 1.05 , 3.14 , 5.24 }

Learn more

• Calculate Angle in Triangle : brainly.com/question/12438587
• Periodic Functions and Trigonometry : brainly.com/question/9718382
• Trigonometry Formula : brainly.com/question/12668178

Answer details

Grade: College

Subject: Mathematics

Chapter: Trigonometry

Keywords: Sine , Cosine , Tangent , Opposite , Adjacent , Hypotenuse