Solve the equation: sin (3x+30°) =√3/2 for 0°≤x≤90°
we all know that sin 60° = √3/2
so, 3x+30 = 60, and x=10°
Since we're dealing with only the first quadrant, there's no other solution.
Work on me to handle it.
I need working out
Well, first of all, I think it's great that you're embracing the mathematical side of life. Now, let's dive into this equation together.
To solve sin(3x + 30°) = √3/2 for 0° ≤ x ≤ 90°, let's start by figuring out the angle for which the sine is √3/2. We know that sin(60°) = √3/2, so we can simplify our equation to:
3x + 30° = 60°
Now, let's subtract 30° from both sides to isolate 3x:
3x = 60° - 30°
3x = 30°
And finally, dividing both sides by 3:
x = 30°/3
So, x = 10°.
Hence, the solution to the equation sin(3x + 30°) = √3/2 for 0° ≤ x ≤ 90° is x = 10°.
To solve the equation sin (3x+30°) = √3/2 for 0° ≤ x ≤ 90°, we can follow the following steps:
Step 1: Identify the inverse trigonometric function.
In this case, the inverse of sin is arcsin or sin^(-1). Therefore, we need to take the arcsin of both sides of the equation to isolate x.
Step 2: Apply the arcsin to both sides of the equation.
arcsin(sin(3x+30°)) = arcsin(√3/2)
Step 3: Simplify the equation.
The arcsin and sin functions are inverses of each other, so they cancel each other out, leaving us with:
3x + 30° = arcsin(√3/2)
Step 4: Solve for x.
To solve for x, we need to isolate it by subtracting 30° from both sides of the equation:
3x = arcsin(√3/2) - 30°
Step 5: Evaluate the expression on the right-hand side.
arcsin(√3/2) is a special angle in the unit circle that corresponds to 60°. Therefore, we can substitute this value into the equation:
3x = 60° - 30°
3x = 30°
Step 6: Solve for x.
Divide both sides of the equation by 3:
x = 30° / 3
x = 10°
Therefore, the solution to the equation sin (3x+30°) = √3/2 for 0° ≤ x ≤ 90° is x = 10°.