1 Given that sin x=0.5,find the values of x between 90° and 360° with solving

2 solve (x-2)(x-3)=12

1 To find the values of x between 90° and 360° where sin x = 0.5, we need to use the unit circle or a trigonometric table.

Step 1: Draw the unit circle. Divide it into four quadrants.

Step 2: In the first quadrant (0° to 90°), sin x is positive. From the unit circle or a trigonometric table, find the angle where sin x = 0.5. In this case, sin x = 0.5 at 30°.

Step 3: In the second quadrant (90° to 180°), sin x is also positive. To find the corresponding angle, subtract the angle found in step 2 from 180°. In this case, x = 180° - 30° = 150°.

Step 4: In the third quadrant (180° to 270°), sin x is negative. To find the corresponding angle, subtract the angle found in step 2 from 180° and then add 180°. In this case, x = 180° - 30° + 180° = 330°.

Step 5: In the fourth quadrant (270° to 360°), sin x is also negative. To find the corresponding angle, subtract the angle found in step 2 from 360°. In this case, x = 360° - 30° = 330°.

So, the values of x between 90° and 360° where sin x = 0.5 are 30°, 150°, and 330°.

2 To solve the equation (x-2)(x-3) = 12, we can use the quadratic formula or factorization.

Method 1: Factorization

Step 1: Expand the left side of the equation:
x^2 - 5x + 6 = 12

Step 2: Rearrange the equation:
x^2 - 5x + 6 - 12 = 0

Step 3: Simplify the equation:
x^2 - 5x - 6 = 0

Step 4: Factorize the quadratic:
(x - 6)(x + 1) = 0

Step 5: Set each factor equal to zero:
x - 6 = 0 or x + 1 = 0

Step 6: Solve for x:
x = 6 or x = -1

So, the solutions to the equation (x-2)(x-3) = 12 are x = 6 and x = -1.

Method 2: Quadratic Formula
Alternatively, we can solve using the quadratic formula.

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -5, and c = -6.

Substituting these values into the formula:
x = (-(-5) ± √((-5)^2 - 4(1)(-6))) / (2(1))
x = (5 ± √(25 + 24)) / 2
x = (5 ± √49) / 2
x = (5 ± 7) / 2

This gives us two solutions:
x = (5 + 7) / 2 = 6
x = (5 - 7) / 2 = -1

So, the solutions to the equation (x-2)(x-3) = 12 are x = 6 and x = -1.

1) To find the values of x between 90° and 360° given that sin x = 0.5, we need to use the inverse sine function (arcsin).

The inverse sine function (arcsin) gives us the angle whose sin value is a given number. In this case, we know that sin x = 0.5, so we can find the value of x by using the arcsin function.

arcsin(0.5) ≈ 30°

Therefore, one value of x between 90° and 360° is 30°. However, we need to find all the values of x within this range.

Since sine is a periodic function, it repeats its values every 360°. Hence, we can add multiples of 360° to find other values of x within the given range.

x = 30° + 360°k, where k is an integer.

Let's find all the possible values of x within the given range:

For k = 1, x = 30° + 360°(1) = 390°
For k = 2, x = 30° + 360°(2) = 750°
For k = 3, x = 30° + 360°(3) = 1110°

Therefore, the values of x between 90° and 360° when sin x = 0.5 are 30°, 390°, 750°, and 1110°.

2) To solve the equation (x-2)(x-3) = 12, we will first expand the equation and then solve for x.

Expanding the equation:

(x - 2)(x - 3) - 12 = 0
x^2 - 3x - 2x + 6 - 12 = 0
x^2 - 5x - 6 = 0

Now, we have a quadratic equation. To solve it, we can either factorize it or use the quadratic formula.

Let's try to factorize it:

(x - 6)(x + 1) = 0

This means that either (x - 6) = 0 or (x + 1) = 0.

If (x - 6) = 0, then x = 6.
If (x + 1) = 0, then x = -1.

So the solution to the equation (x-2)(x-3) = 12 is x = 6 and x = -1.

you know that sin30° = 1/2

Use that as your reference angle and recall where sinx > 0

(x-2)(x-3) = 12
well, 4*3 = 12, right?
so, x=6
also, -3 * -4 = 12
so, x = -1
Or, if you must,
x^2-5x+6 = 12
x^2-5x-6 = 0
(x-6)(x+1) = 0