CALTRX

Trigonometry Calculator

Calculate sine, cosine, tangent, and their inverses for any angle in degrees or radians. A core tool for math and engineering.

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Tips & Notes

  • ✓SOH-CAH-TOA: Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj. The most important memory device in trig.
  • ✓sin²θ+cos²θ=1 always. Use to find one function from another without knowing the angle.
  • ✓Always check calculator mode: degrees or radians. sin(90°)=1 but sin(90 radians)≈0.894.
  • ✓tan θ = sin θ/cos θ. Undefined when cos θ=0 (at 90° and 270°).
  • ✓For reference angles: sin and cos are positive in Q1, sin positive in Q2, tan in Q3, cos in Q4.

Common Mistakes

  • ✗Calculator in wrong mode (degrees vs radians). Always verify mode before computing trig values.
  • ✗Confusing inverse trig with reciprocal: arcsin(x) ≠ 1/sin(x). arcsin is the inverse function.
  • ✗Thinking sin(2θ)=2sin(θ). Correct: sin(2θ)=2sin(θ)cos(θ) — double angle formula.
  • ✗Mixing up opposite and adjacent sides when setting up ratios — opposite is across from angle θ.
  • ✗Forgetting negative signs in other quadrants: sin is negative in Q3 and Q4.

Trigonometry Calculator Overview

Trigonometry studies the relationships between the angles and side lengths of triangles, and extends these relationships to all angles through the unit circle. The six trigonometric functions — sine, cosine, tangent and their reciprocals — encode geometric relationships that appear in wave physics, electrical engineering, navigation, signal processing, and computer graphics. Every oscillating system from sound waves to electromagnetic radiation is described by sine and cosine functions.

The six trigonometric functions for angle θ in a right triangle (O=opposite, A=adjacent, H=hypotenuse):

sin θ = O/H | cos θ = A/H | tan θ = O/A | csc θ = H/O | sec θ = H/A | cot θ = A/O
SOH-CAH-TOA: the memory device for the three primary ratios. For the angle opposite a side of 3 in a 3-4-5 triangle: sin=3/5=0.6, cos=4/5=0.8, tan=3/4=0.75. Key angle values to memorize:
0°: sin=0, cos=1, tan=0 | 30°: sin=1/2, cos=√3/2, tan=1/√3 | 45°: sin=√2/2, cos=√2/2, tan=1
60°: sin=√3/2, cos=1/2, tan=√3 | 90°: sin=1, cos=0, tan=undefined
Pythagorean identity — the most important trigonometric identity:
sin²θ + cos²θ = 1 for all angles θ
EX: If sin θ = 0.6, then cos θ = √(1−0.36) = √0.64 = 0.8 | tan θ = sin/cos = 0.6/0.8 = 0.75
From sin²+cos²=1: dividing by cos² gives tan²+1=sec². Dividing by sin² gives 1+cot²=csc². Inverse trigonometric functions find angles from ratios:
EX: sin θ = 0.5 → θ = arcsin(0.5) = 30° | tan θ = 1 → θ = arctan(1) = 45°
Ranges: arcsin output: [−90°,90°]. arccos output: [0°,180°]. arctan output: (−90°,90°). Law of Sines (for any triangle, not just right): a/sin A = b/sin B = c/sin C. Law of Cosines (extends Pythagorean theorem): c² = a²+b²−2ab·cos C. Angle mode warning: sin(90) in DEGREE mode = 1.0 exactly. sin(90) in RADIAN mode ≈ 0.8940. Always verify calculator mode before computing. In calculus, radians are required — d/dx[sin(x)]=cos(x) only when x is in radians. Real-world applications: Navigation uses bearing angles computed with arctan. Architecture: a roof with 4:12 pitch has angle arctan(4/12)=18.4°. Electrical engineering: AC voltages and currents are sinusoidal — V(t)=V₀sin(ωt+φ). Physics: projectile trajectory has horizontal component v·cos(θ) and vertical v·sin(θ). Sound, light, and radio waves all follow sinusoidal equations. Trigonometric functions extend naturally beyond the 0–90° range of right triangles through the unit circle definition. For any angle θ, place a point on the unit circle (radius = 1) at angle θ from the positive x-axis.

Frequently Asked Questions

Sine, cosine, and tangent relate angles to side ratios in right triangles. For angle θ in a right triangle: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. Memory aid: SOH-CAH-TOA. Example: right triangle with hypotenuse 10, angle 30°. Opposite = 10 × sin(30°) = 10 × 0.5 = 5. Adjacent = 10 × cos(30°) = 10 × (√3/2) ≈ 8.66. These ratios hold for any right triangle with the same angle, regardless of size.

The Pythagorean identity sin²(θ) + cos²(θ) = 1 holds for all angles. This comes directly from the Pythagorean theorem applied to the unit circle (radius = 1): opposite² + adjacent² = hypotenuse² = 1. From this fundamental identity: tan²(θ) + 1 = sec²(θ) and 1 + cot²(θ) = csc²(θ). These identities simplify trigonometric expressions and are used constantly in integration, proof, and physics — especially when converting between trig functions.

For angles in any quadrant, determine the sign of each trig function using ASTC (All Students Take Calculus): All functions positive in Q1, Sine positive in Q2, Tangent positive in Q3, Cosine positive in Q4. Example: sin(120°) — 120° is in Q2 where sine is positive. Reference angle = 180°−120° = 60°. sin(120°) = +sin(60°) = +√3/2 ≈ 0.866. cos(120°) — Q2, cosine is negative: cos(120°) = −cos(60°) = −1/2. Always find the reference angle first, then apply the sign rule.

The Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). The Law of Cosines: c² = a² + b² − 2ab×cos(C). Use Law of Sines when you know two angles and one side (AAS or ASA) or two sides and the angle opposite one of them (SSA — check for ambiguous case). Use Law of Cosines when you know two sides and the included angle (SAS) or all three sides (SSS). The Law of Cosines reduces to the Pythagorean theorem when C = 90°, since cos(90°) = 0.

Inverse trig functions find the angle when you know the ratio. sin⁻¹(0.5) = 30° — the angle whose sine is 0.5. cos⁻¹(√3/2) = 30°. tan⁻¹(1) = 45°. Ranges: sin⁻¹ returns −90° to 90°, cos⁻¹ returns 0° to 180°, tan⁻¹ returns −90° to 90°. Multiple angles can have the same trig value — inverse trig returns only the principal value. Example: both 30° and 150° have sin = 0.5, but sin⁻¹(0.5) = 30° only. Use context to determine whether the other solution is also valid.

Trig functions appear in signal processing as Fourier series: any periodic signal decomposes into sums of sines and cosines at different frequencies and amplitudes. A square wave of frequency f: f(t) = (4/π)[sin(2πft) + sin(6πft)/3 + sin(10πft)/5 + ...]. The Fourier transform generalizes this to non-periodic signals. Audio equalization, image compression (JPEG uses cosine transforms), and wireless communication all rely on this trigonometric decomposition. Physics uses sin and cos for wave equations, AC circuits, and orbital mechanics.