Solve 4sin(x)² = 3
Trigonometric equation, worked out line by line the way a teacher would write it.
Answer
| General solution | x = π/3 + 2π·n; x = 2π/3 + 2π·n; x = 4π/3 + 2π·n; x = 5π/3 + 2π·n (n any integer) |
Step-by-step solution
15 steps-
1 Given4 \sin^{2}{\left(x \right)} = 3
Solve for x.
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2 Substitute t = sin(x)
Treat sin(x) as a single unknown t: the equation becomes an ordinary polynomial in t. Solve for t, then find the angles that have each value.
4 t^{2} - 3 = 0 -
3 This is a quadratic in standard form at² + bt + c = 0
Every quadratic equation can be arranged as at² + bt + c = 0. Reading off a = 4, b = 0 and c = -3, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 4,\quad b = 0,\quad c = -3 -
4 There is no t term, so isolate t²t^{2} = \frac{3}{4}
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5 Take the square root of both sides — remember both signs
A number and its negative give the same result when raised to an even power: 3² = 9 and (−3)² = 9. So if something squared equals k, that something is √k or −√k. Forgetting the minus sign loses a solution.
t = \pm \frac{\sqrt{3}}{2} -
6 Solutionst = \frac{\sqrt{3}}{2}\quad \text{or} \quad t = - \frac{\sqrt{3}}{2}
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7 Back-substitute: sin(x) = √3/2\sin{\left(x \right)} = \frac{\sqrt{3}}{2}
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8 Reference angle: the acute angle α with sin α = √3/2
The reference angle α is the acute angle (between 0 and 90°) whose sine, cosine or tangent has this value, ignoring the sign. The other angles with the same value are built from it in the next step.
\alpha = \arcsin\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \approx 1.04719758 -
9 Sine is positive in quadrants I and II: u = α or u = π − α
In one full turn, sine and cosine reach each value between −1 and 1 twice. The signs follow the quadrants: everything is positive in quadrant I, only sine in II, only tangent in III and only cosine in IV.
x = \frac{\pi}{3}\quad \text{or} \quad x = \frac{2 \pi}{3} -
10 Back-substitute: sin(x) = -√3/2\sin{\left(x \right)} = - \frac{\sqrt{3}}{2}
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11 Reference angle: the acute angle α with sin α = √3/2\alpha = \arcsin\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \approx 1.04719758
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12 Sine is negative in quadrants III and IV: u = π + α or u = 2π − αx = \frac{4 \pi}{3}\quad \text{or} \quad x = \frac{5 \pi}{3}
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13 General solution (n is any integer)
Sine and cosine repeat every full turn (360° or 2π) and tangent every half turn, so adding any whole number n of periods gives another solution. n can be any integer, positive, negative or 0.
x = \frac{\pi}{3} + 2 \pi n\quad \text{or} \quad x = \frac{2 \pi}{3} + 2 \pi n\quad \text{or} \quad x = \frac{4 \pi}{3} + 2 \pi n\quad \text{or} \quad x = \frac{5 \pi}{3} + 2 \pi n,\quad n \in \mathbb{Z} -
14 The solutions in one turn, 0 ≤ x < 2πx = \frac{\pi}{3},\; \frac{2 \pi}{3},\; \frac{4 \pi}{3},\; \frac{5 \pi}{3}
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15 Check\begin{aligned}x = \frac{\pi}{3}:\quad 3 = 3\quad\checkmark\\ x = \frac{2 \pi}{3}:\quad 3 = 3\quad\checkmark\\ x = \frac{4 \pi}{3}:\quad 3 = 3\quad\checkmark\\ x = \frac{5 \pi}{3}:\quad 3 = 3\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| pi/3 | 3 | 3 | ✓ |
| 2pi/3 | 3 | 3 | ✓ |
| 4pi/3 | 3 | 3 | ✓ |
| 5pi/3 | 3 | 3 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.