\sin \theta = \frac{\sqrt2}2 \quad \Rightarrow \quad \theta = \frac\pi4 \quad \textor \quad \theta = \frac3\pi4

Understanding θ in the Equation sin θ = √2/2: Solutions θ = π/4 and θ = 3π/4

When studying trigonometric functions, one of the most fundamental and frequently encountered equations is:
sin θ = √2 / 2

This simple equation carries profound meaning in both mathematics and applied fields, as it identifies key angles where the sine function reaches a specific value. For those new to trigonometry or revisiting these concepts, we’ll explore what this equation means, how to solve it, and why θ = π/4 and θ = 3π/4 are critical solutions.

Understanding the Context

What Does sin θ = √2/2 Represent?

The sine function gives the ratio of the opposite side to the hypotenuse in a right triangle, or more generally, in the unit circle, it represents the vertical coordinate (y-coordinate) of a point at angle θ.

The value √2 / 2, approximately 0.707, appears repeatedly in standard angles due to its exact value on the unit circle. Specifically, this value corresponds to 45° and 135° — angles measured in radians as π/4 and 3π/4, respectively.

$\sin \theta = \frac{\sqrt{2}}{2} \quad \Rightarrow \quad \theta = \frac{\pi}{4} \quad \text{or} \quad \theta = \frac{3\pi}{4}$

Key Insights

Solving sin θ = √2 / 2: Step-by-Step

To solve sin θ = √2 / 2, we use key knowledge about the sine function’s behavior:

Step 1: Recall exact values on the unit circle

On the unit circle, sine values equal √2 / 2 at two key angles:

sin(π/4) = √2 / 2 (45° in the first quadrant, where both x and y are positive)
sin(3π/4) = √2 / 2 (135° in the second quadrant, where sine is positive but cosine is negative)

Step 2: Use the unit circle symmetry

The sine function is positive in both the first and second quadrants. Thus, there are two solutions within one full rotation (0 ≤ θ < 2π):

θ = π/4 — first quadrant
θ = 3π/4 — second quadrant

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Final Thoughts

Step 3: General solution (optional)

Because sine is periodic with period 2π, the complete solution set is:
θ = π/4 + 2πn or θ = 3π/4 + 2πn, where n is any integer.
This captures every angle where sine equals √2 / 2 across the number line.

Why Are These the Only Solutions in [0, 2π)?

Within the standard interval from 0 to 2π (a full circle), sine reaches √2 / 2 only at those two angles due to symmetry and the monotonicity of sine in key intervals:

In 0 to π/2 (0 to 90°): only π/4 gives sin θ = √2 / 2
In π/2 to π (90° to 180°): only 3π/4 gives the correct sine value
Beyond π, sine values decrease or change sign, never again hitting √2 / 2 exactly until the next cycle

This exclusive pair ensures accuracy in solving trigonometric equations and modeling periodic phenomena like waves, motion, and oscillations.

Real-World Applications

Understanding these solutions isn’t just theoretical:

Engineering: Used in signal processing and AC circuit analysis
Physics: Essential for analyzing wave interference, pendulum motion, and rotational dynamics
Navigation & Geometry: Helps determine directional angles and coordinate transformations
Computer Graphics: Enables accurate rotation and periodic motion in animations