### Solving Trigonometric Equations

**E.g.1**

Solve sin x = 0.5 for 0 ≤ x ≤ 360

sin x = 0.5

x = 30^{0}

Since y = 0.5 line crosses the sine curve at two points, there are two solutions.

Now, look at the symmetry of the graph; the two values of x are 30^{0} and 150^{0}.

x = 30 and 150.

**E.g.2**

Solve cos x = -0.5 for 0 ≤ x ≤ 360

cos x = -0.5

x = 120^{0}

Since y = -0.5 line crosses the sine curve at two points, there are two solutions.

Now, look at the symmetry of the graph; the two values of x are 120^{0} and 240^{0}.

x = 120 and 240.

**E.g.3**

Solve sin (x +10) = 0.5 for 0 ≤ x ≤ 360

From example 1,

(x + 10) = 30^{0}

There are two values for (x +10) that satisfy the equation; they are 30^{0} and 150^{0}

x + 10 = 30 or x + 10 = 150

x = 20 or x = 140

**E.g.4**

Solve 1 + 2 sin x = 2 for 0 ≤ x ≤ 360

2 sin x = 1

sin x = 0.5

From example 1,

x = 30

^{0}
There are two values for x that satisfy the equation; they are 30

^{0} and 150

^{0}
x = 30 or x = 150.

#### Solving Equations - interactive

You can animate the y=a line by clicking the *play* button or move the slider to a point you want it to be at.

**Now, in order to complement what you have just learnt, work out the following questions:**

- Solve 2 sin x = 1.5 for 0 ≤ x ≤ 360
- Solve 1 + 3 cos x = 2 for 0 ≤ x ≤ 360
- Solve sin (2x - 10) = 0.7 for 0 ≤ x ≤ 360
- Solve tan (x -30) = 0.7 for 0 ≤ x ≤ 360
- Solve cos (2x -20) = 0.5 for 0 ≤ x ≤ 360
- Solve sin
^{2} x = 0.25 for 0 ≤ x ≤ 360
- Solve sin 2x = 0.5 for 0 ≤ x ≤ 360
- Solve 1 + 3 sin 2x = 2 for 0 ≤ x ≤ 360