Calculate trigonometric functions, inverse functions, and solve triangles
Trigonometry is the study of relationships between angles and sides of triangles. The word comes from Greek: 'trigonon' (triangle) and 'metron' (measure). It's fundamental to mathematics, physics, engineering, astronomy, and many other fields.
The six trigonometric functions - sine, cosine, tangent, cosecant, secant, and cotangent - relate an angle to ratios of sides in a right triangle. These functions extend beyond triangles to describe periodic phenomena like waves, oscillations, and circular motion.
Trigonometry connects geometry to algebra and enables solutions to problems involving angles, distances, and periodic behavior. From calculating building heights to modeling sound waves, trigonometry is everywhere.
Opposite / Hypotenuse. Primary function for waves and oscillations.
Adjacent / Hypotenuse. 90° phase shift from sine.
Opposite / Adjacent = sin/cos. Slope of a line at angle θ.
csc = 1/sin, sec = 1/cos, cot = 1/tan.
Memorize these special angle values for quick reference:
| Angle | 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 |
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ. Derived from the Pythagorean theorem.
sin(A±B) = sinA·cosB ± cosA·sinB. cos(A±B) = cosA·cosB ∓ sinA·sinB. Essential for combining angles.
sin(2θ) = 2sinθ·cosθ. cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ.
sin(90°-θ) = cosθ, cos(90°-θ) = sinθ. Functions of complementary angles are related.
Use degrees for everyday angles (architecture, navigation). Use radians for calculus, physics, and advanced math - they simplify formulas. π radians = 180°.
tan(90°) = sin(90°)/cos(90°) = 1/0. Division by zero is undefined. The tangent function has vertical asymptotes at 90°, 270°, etc.
arcsin, arccos, arctan (also written sin⁻¹, cos⁻¹, tan⁻¹) find the angle given a ratio. If sin(30°) = 0.5, then arcsin(0.5) = 30°.
Use Law of Sines (a/sinA = b/sinB = c/sinC) when you know an angle and its opposite side. Use Law of Cosines (c² = a² + b² - 2ab·cosC) for other cases.
A right triangle has a 30° angle and a hypotenuse of 10 units. Find the side opposite the 30° angle and the adjacent side.
Resultopposite = 5, adjacent = 5√3 ≈ 8.66
Use SOH-CAH-TOA. sin(30°) = opp/hyp, so opp = 10 × sin(30°) = 10 × 0.5 = 5. cos(30°) = adj/hyp, so adj = 10 × cos(30°) = 10 × (√3/2) ≈ 8.660. Verify with the Pythagorean theorem: 5² + (5√3)² = 25 + 75 = 100 = 10². The 30-60-90 triangle always has side ratios 1 : √3 : 2.
It is a mnemonic for the three primary trig ratios in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. The hypotenuse is the side opposite the right angle; opposite and adjacent are named relative to the angle you are solving for.
Use the Law of Sines (a/sinA = b/sinB = c/sinC) when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). Use the Law of Cosines (c² = a² + b² − 2ab·cosC) when you know three sides (SSS) or two sides and the included angle (SAS), where the Law of Sines alone cannot start the solution.
Degrees divide a full circle into 360 equal parts; radians measure angle by arc length on the unit circle, so a full circle is 2π radians. Convert with radians = degrees × π/180. Calculus and most physics formulas assume radians, while construction, navigation, and everyday geometry typically use degrees. Always check your calculator's mode before evaluating sin, cos, or tan.
Inverse functions arcsin, arccos, and arctan (written sin⁻¹, cos⁻¹, tan⁻¹) return the angle that produces a given ratio. For example, arcsin(0.5) = 30° because sin(30°) = 0.5. Each inverse has a restricted range to remain a function: arcsin and arctan return values in [−90°, 90°], while arccos returns values in [0°, 180°].
The unit circle is a circle of radius 1 centered at the origin. For any angle θ measured counterclockwise from the positive x-axis, the point on the circle has coordinates (cos θ, sin θ). This definition extends sin and cos to all real angles, explains their signs in each quadrant, and shows why both functions are periodic with period 360° (2π radians).
For 30°, 45°, 60°, and 90°, memorize sin and cos values: sin(30°) = 1/2, cos(30°) = √3/2; sin(45°) = cos(45°) = √2/2; sin(60°) = √3/2, cos(60°) = 1/2; sin(90°) = 1, cos(90°) = 0. Tangent is sine divided by cosine, so tan(45°) = 1 and tan(90°) is undefined. These values appear constantly in geometry, physics, and engineering problems.
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