Trigonometry Calculator

**Six trig functions** (for angle θ in a right triangle): - sin(θ) = opposite / hypotenuse - cos(θ) = adjacent / hypotenuse - tan(θ) = opposite / adjacent = sin / cos - csc(θ) = 1 / sin(θ) - sec(θ) = 1 / cos(θ) - cot(θ) = 1 / tan(θ) = cos / sin **Inverse functions:** - arcsin(y) = θ such that sin(θ) = y. Range: [-90°, 90°]. - arccos(y) = θ such that cos(θ) = y. Range: [0°, 180°]. - arctan(y) = θ such that tan(θ) = y. Range: [-90°, 90°]. **Common angles and values:** | Angle (°) | Angle (rad) | sin | cos | tan | |---|---|---|---|---| | 0 | 0 | 0 | 1 | 0 | | 30 | π/6 | 1/2 | √3/2 | 1/√3 | | 45 | π/4 | √2/2 | √2/2 | 1 | | 60 | π/3 | √3/2 | 1/2 | √3 | | 90 | π/2 | 1 | 0 | undefined | | 180 | π | 0 | -1 | 0 | | 270 | 3π/2 | -1 | 0 | undefined | | 360 | 2π | 0 | 1 | 0 | **Pythagorean identity:** sin²(θ) + cos²(θ) = 1 For any angle. Derived from Pythagorean theorem on unit circle. **Worked example:** For θ = 30°: sin(30°) = 0.5 cos(30°) = √3/2 ≈ 0.866 tan(30°) = sin/cos = 0.5 / 0.866 ≈ 0.577 Verify: sin² + cos² = 0.25 + 0.75 = 1 ✓. **Worked example: inverse** arcsin(0.5) = 30°. arccos(0.5) = 60°. arctan(1) = 45°. **Degrees vs radians:** Convert: - Radians to degrees: × 180/π. - Degrees to radians: × π/180. | Degrees | Radians | |---|---| | 30 | π/6 | | 45 | π/4 | | 60 | π/3 | | 90 | π/2 | | 180 | π | | 360 | 2π | Most programming languages use radians; calculators have degree/radian mode toggle. **Trig identities:** | Identity | Statement | |---|---| | Pythagorean | sin²θ + cos²θ = 1 | | Reciprocal | csc = 1/sin, sec = 1/cos, cot = 1/tan | | Quotient | tan = sin/cos, cot = cos/sin | | Even/Odd | sin(-θ) = -sin(θ); cos(-θ) = cos(θ); tan(-θ) = -tan(θ) | | Periodicity | sin/cos: period 2π; tan: period π | | Sum formulas | sin(A+B) = sin A cos B + cos A sin B | | Difference | sin(A-B) = sin A cos B - cos A sin B | | Cosine sum | cos(A+B) = cos A cos B - sin A sin B | | Double angle | sin(2θ) = 2 sin θ cos θ | | | cos(2θ) = cos²θ - sin²θ | **Unit circle:** Circle of radius 1 centered at origin. For angle θ (measured from positive x-axis): Point on circle: (cos θ, sin θ). This visualizes trig functions for any angle. **Quadrants:** | Quadrant | Angle range | sin | cos | tan | |---|---|---|---|---| | I | 0-90° | + | + | + | | II | 90-180° | + | - | - | | III | 180-270° | - | - | + | | IV | 270-360° | - | + | - | Mnemonic: "All Students Take Calculus" (all positive in I; sin in II; tan in III; cos in IV). **Right triangle ratios:** For right triangle with hypotenuse 1: - sin(angle) = opposite leg - cos(angle) = adjacent leg For non-unit hypotenuse: - sin × hypotenuse = opposite leg - cos × hypotenuse = adjacent leg - tan = opposite/adjacent **Solving triangles:** For right triangle, given one acute angle θ and hypotenuse h: - Opposite leg = h × sin(θ). - Adjacent leg = h × cos(θ). For right triangle, given two legs: - tan(angle) = opposite / adjacent. - angle = arctan(opposite/adjacent). **Worked example: surveying** Surveyor measures angle 30° to top of tree from 50 m away. Tree height? tan(30°) = height / 50. height = 50 × tan(30°) ≈ 50 × 0.577 ≈ 28.87 m. **Worked example: navigation** Ship sails 100 km at heading 40° from north. North/east distance? East = 100 × sin(40°) ≈ 100 × 0.643 = 64.3 km. North = 100 × cos(40°) ≈ 100 × 0.766 = 76.6 km. **Periodic functions:** sin and cos are periodic with period 2π (360°): sin(θ + 360°) = sin(θ). cos(θ + 360°) = cos(θ). tan has period π (180°): tan(θ + 180°) = tan(θ). This periodicity makes them fundamental for modeling waves, oscillations, rotations. **Common applications:** - **Navigation**: bearings, GPS, dead reckoning. - **Surveying**: indirect measurement. - **Astronomy**: positional astronomy, parallax. - **Architecture**: roof angles, structural design. - **Engineering**: rotational analysis, mechanical design. - **Physics**: waves, oscillations, projectile motion. - **Computer graphics**: rotations, transformations. - **Audio**: signal processing, Fourier analysis. - **AC electricity**: voltage and current oscillate sinusoidally. - **Music**: frequencies, intervals as ratios. **Trigonometry in physics:** - **Projectile motion**: x = v₀cos(θ)t, y = v₀sin(θ)t - (1/2)gt². - **Simple harmonic motion**: x(t) = A cos(ωt + φ). - **AC voltage**: V(t) = V_peak × sin(2πft + φ). - **Wave equations**: y(x,t) = A sin(kx - ωt). Trigonometry essential for understanding nearly all wave/oscillation phenomena. **Common identities to know:** sin(x + y) = sin x cos y + cos x sin y. cos(x + y) = cos x cos y - sin x sin y. tan(x + y) = (tan x + tan y) / (1 - tan x tan y). Double angle: sin(2x) = 2 sin x cos x. cos(2x) = cos²x - sin²x = 1 - 2sin²x = 2cos²x - 1. Half angle: sin²(x/2) = (1 - cos x) / 2. cos²(x/2) = (1 + cos x) / 2. **Inverse trig ranges:** - arcsin: [-π/2, π/2] or [-90°, 90°]. - arccos: [0, π] or [0°, 180°]. - arctan: (-π/2, π/2) or (-90°, 90°). Calculator gives principal value; for full solution, add periods. **Software:** - **Calculators**: sin, cos, tan buttons (check degree/radian mode). - **Python (math)**: math.sin, math.cos, math.tan (radians). - **JavaScript**: Math.sin, Math.cos, Math.tan (radians). - **Excel**: =SIN(RADIANS(angle)) to use degrees. - **Wolfram Alpha**: handles all forms. **Programming radians vs degrees:** Most programming languages use radians. Convert: - Degrees: math.degrees(rad) or rad × 180/math.pi. - Radians: math.radians(deg) or deg × math.pi/180. Be careful: forgetting conversion is common bug. **Common applications:** - **Architecture**: roof slope, structural angles. - **Engineering**: rotational machinery, optics. - **Surveying**: triangulation, elevation. - **Navigation**: GPS, marine, aviation. - **Astronomy**: orbital calculations. - **Photography**: field of view. - **Sports analytics**: trajectory analysis. - **Manufacturing**: angled cuts, gear design. **Pitfalls:** - **Degree vs radian confusion**: most programming uses radians. - **Quadrant for inverse**: arctan gives only one of two possible angles. - **Undefined values**: tan(90°), tan(270°), etc. - **Wrong identity**: many trig identities; memorize key ones. - **Sign in quadrant**: trig signs depend on quadrant. - **Periodic ambiguity**: sin(30°) = sin(150°), etc. **Educational notes:** Trigonometry taught in: - 8th-9th grade: introduction with right triangles. - High school trigonometry: unit circle, identities, equations. - Pre-calculus: graphs, transformations. - Calculus: derivatives, integrals of trig functions. - Physics: applications. - Engineering: throughout. Foundation for advanced math and physical sciences. **Pitfalls (continued):** - **For very small angles**: sin θ ≈ θ (radians), useful approximation. - **For 90° complements**: sin and cos swap. - **For special angles**: memorize exact values. - **For calculators**: ensure mode (degree/radian) matches problem.