Let us understand how sin2x works

The trigonometric identity is the identity of angles. The trigonometric identity is also called as circular identity. They are used for relating the angles of a triangle to the lengths of the sides of a triangle. The trigonometric identity is essential for the basic study of triangles and representation of periodic phenomena, surrounded by many other applications. One of the trigonometric identity is the sine identity.Question: Solve, sin2x × sin x = 0, to get the angle value of x from the double angle.

Solution: Given that, sin2x × sin x = 0.
We know that, sin2θ = 2sinθ cosθ,
=> sin2θ = 2sinθ cosθ,
Now plug the value of sin2x in the equation sin2x + sin x = 0, we get,
=> (2sin x cos x) × sin x = 0,
=> 2sin x cos x × sin x = 0,
=> 2sin²x × cos x= 0,
Split the given expression into two terms, we get,
=> 2sin²x = 0, and sin x = 0,
=> sin²x = 0, and sin x = 0,
=> sin x = 0, and sin x = 0,
=> x = sin⁻¹(0), and x = cos⁻¹(0),

=> x = 0° (or) 360°, and x = 90°,

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Simultaneous Equations

Definition of Simultaneous Equations

 A pair of "Simultaneous equations" is two equations which are both true at the same time. You have two equations which have two unknowns to be found.

Example

 A man buys 3 fish and 2 chips for £2.80

A woman buys 1 fish and 4 chips for £2.60

How much are the fish and how much are the chips?

 First we form the equations. Let fish be f and chips be c.

We know that:

3f + 2c = 280    (1)

f + 4c = 260      (2)

 There are two methods of solving simultaneous equations. Use the method which you prefer:

Elimination

 This involves changing the two equations so that one can be added/ subtracted from the other to leave us with an equation with only one unknown (which we can solve). We can 'change' the equations by multiplying them through by a constant- as long as we multiply both sides of the equation by the same number it will remain true.

 In our above example:

 Doubling (1) gives:

6f + 4c = 560 (3)

 Since equation (2) has a 4c in it, we can subtract this from the new equation (3) and the c's will all have disappeared:

(3)-(2) gives 5f = 300  ∴ f = 60

Therefore the price of fish is 60p

 So we can put f=60 in either of our original equations. Substitute this value into (1): 3(60) + 2c = 280 ∴ 2c = 100

c = 50

Therefore the price of chips is 50p

Substitution

 The method of substitution involves transforming one equation into x = (something) or y = (something) and then substituting this something into the other equation.

 So,Rearrange one of the original equations to isolate a variable.

Rearranging (2): f = 260 - 4c

Substitute this into the other equation:3(260 - 4c) + 2c = 280∴ 780 - 12c + 2c = 280 ∴ 10c = 500

∴ c = 50

Substitute this into one of the original equations to get f = 60 .

 Harder simultaneous equations

 To solve a pair of equations, one of which contains x2, y2 or xy, we need to use the method of substitution.

Example

 2xy + y = 10  (1)

x + y = 4        (2)

Take the simpler equation and get y = .... or x = ....

from (2), y = 4 - x    (3)

this can be substituted in the first equation. Since y = 4 - x, where there is a y in the first equation, it can be replaced by 4 - x .

sub (3) in (1), 2x(4 - x) + (4 - x) = 10

∴ 8x - 2x2 + 4 - x - 10 = 0

∴ 7x - 2x2 - 6 = 0

∴ 2x2 - 7x + 6 = 0  (taking everything to the other side of the equals sign)

∴ (2x - 3)(x - 2) = 0

∴ either 2x - 3 = 0 or x - 2 = 0

therefore x = 1.5 or 2 .

 Substitute these x values into one of the original equations.

When x = 1.5,  y = 2.5

when x = 2, y = 2

Using Graphs

 You can solve simultaneous equations by drawing graphs of the two equations you wish to solve. The x and y values of where the graphs intersect are the solutions to the equations.

Example

 Solve the simultaneous equations 3y = -2x + 6 and y = 2x -2 by graphical methods.

  

  From the graph, y = 1 and x = 1.5 (approx.). These are the answers to the simultaneous equations.