Understanding Sine, Arcsine, and Their Unit‑Circle Foundations

Introduction

The sine function and its inverse, arcsine, are fundamental in trigonometry. This article walks through how to evaluate sin(π/4), derive its exact value using a 45‑45‑90 triangle, and understand the arcsine function, its domain, range, and how to compute specific values such as arcsin(‑√3/2).

Sine of π/4

• Radians vs. degrees: π/4 radians equals 45°.
• Unit‑circle definition: The sine of an angle is the y‑coordinate of the point where the terminal side of the angle intersects the unit circle.
• Visual cue: At π/4 the point lies on the line y = x in the first quadrant.

Deriving the Value Using a 45‑45‑90 Triangle

1. Draw a right triangle with the hypotenuse equal to the unit‑circle radius 1.
2. Because the base angles are both 45°, the legs are equal (let each be x).
3. Apply the Pythagorean theorem: - x² + x² = 1² → 2x² = 1 → x² = ½ → x = √½ = 1/√2.
4. Rationalise the denominator: 1/√2 = √2/2.
5. The y‑coordinate (and thus sin π/4) is √2/2.

What Is Arcsine?

• Definition: arcsin (or inverse sine) asks, “Which angle has a given sine value?”.
• Notation: arcsin(y) = θ ⇔ sin θ = y.
• Example: Since sin π/4 = √2/2, arcsin(√2/2) = π/4.

Domain and Range of the Inverse Sine

• Domain: Because sine outputs values only between ‑1 and 1, the input to arcsin must satisfy ‑1 ≤ x ≤ 1.
• Range restriction: To make arcsin a true function (one output per input), its range is limited to the interval [‑π/2, π/2] (first and fourth quadrants).

Example: arcsin(‑√3/2)

1. Recognise √3/2 as the y‑coordinate of a 30‑60‑90 triangle on the unit circle.
2. The positive √3/2 corresponds to a 60° (π/3) angle in the first quadrant; the negative sign places the point in the fourth quadrant.
3. Therefore the angle is ‑60° or ‑π/3 radians.
4. Verify with a calculator set to radian mode: - arcsin(‑√3/2) ≈ ‑1.047 rad, which matches ‑π/3.

Verifying with a Calculator

• Switch the calculator to radian mode.
• Use the inverse sine (often labelled sin⁻¹) function.
• Input the value (e.g., ‑√3/2) and compare the output to the exact fraction of π.

Key Points to Remember

• Sine = y‑coordinate on the unit circle.
• 45‑45‑90 triangle yields sin π/4 = √2/2.
• Arcsine returns the unique angle in [‑π/2, π/2] whose sine equals the given value.
• Domain of arcsin: [‑1, 1]; Range: [‑π/2, π/2].
• Use the unit‑circle geometry to find exact values for common angles.

The sine of π/4 is √2/2, and the arcsine function provides the unique angle within [‑π/2, π/2] that produces a given sine value; understanding the unit‑circle geometry and the domain‑range restrictions makes evaluating both functions straightforward.