Trigonometry Calculator — Sine, Cosine & Tangent

Trigonometry Guide

What's the key thing to understand about SOH CAH TOA?

In a right triangle: sin θ = Opposite/Hypotenuse. cos θ = Adjacent/Hypotenuse. tan θ = Opposite/Adjacent. Finding a side: multiply the known side by the trig ratio. Finding an angle: use inverse trig (arcsin, arccos, arctan). Example: right triangle with hypotenuse 10, angle 30°: opposite = 10 × sin 30° = 10 × 0.5 = 5. adjacent = 10 × cos 30° = 10 × 0.866 = 8.66. Special angles: sin 30° = 0.5, sin 45° = √2/2 ≈ 0.707, sin 60° = √3/2 ≈ 0.866. These should be memorised — exact values are required in GCSE and A-level exams, where using a calculator approximation instead of the exact surd form can cost marks.

What should I know about Sine Rule?

a/sin A = b/sin B = c/sin C. Used when: two angles and one side known (AAS or ASA). Two sides and a non-included angle known (SSA — ambiguous case). Finding side: a = b × sin A / sin B. Finding angle: sin A = a × sin B / b. Example: a=5, angle A=40°, angle B=70°. b = 5 × sin 70° / sin 40° = 5 × 0.940 / 0.643 = 7.31. Ambiguous case (SSA): two triangles may be possible when the given angle is acute and the opposite side is shorter than the adjacent side.

What do I need to know about Cosine Rule?

Finding a side: a² = b² + c² − 2bc cos A. Finding angle: cos A = (b² + c² − a²) / (2bc). Used when: three sides known (SSS). Two sides and included angle known (SAS). Example (SAS): b=5, c=8, angle A=60°. a² = 25 + 64 − 2×5×8×cos60° = 89 − 40 = 49. a = 7. Triangle area: Area = ½ab sin C (any two sides and included angle).

What's the key thing to understand about Graphs and Identities?

Key identities: sin²θ + cos²θ = 1 (Pythagorean identity). tan θ = sin θ / cos θ. Graphs: sin and cos repeat with period 360°. Range: −1 to +1. tan repeats with period 180°, with asymptotes at ±90°. Amplitude of y = a sin(bx + c) + d: amplitude = |a|. Period = 360°/|b|. Phase shift = −c/b. Transformations of trig graphs are commonly examined at GCSE and A-level — understanding amplitude, period, and phase shift from the equation is essential.

What should I know about The Unit Circle?

The unit circle (radius 1, centred at the origin) defines sine and cosine for any angle, not just the acute angles that fit inside a right triangle. For an angle θ measured from the positive x-axis, cos(θ) is the x-coordinate and sin(θ) is the y-coordinate of the point on the circle. This extends trig functions beyond 0-90°: sin(150°) = sin(180°−150°) = sin(30°) = 0.5, because 150° and 30° are reflections across the y-axis giving the same height. Similarly cos(120°) = −cos(60°) = −0.5, negative because 120° places the point in the second quadrant where x-coordinates are negative. Remembering which quadrants give positive or negative sine, cosine, and tangent (often taught as 'CAST' or 'SOH CAH TOA all positive in quadrant 1') lets you find exact values for angles like 120°, 210°, or 300° using only the 0-90° reference values you already know.

What do I need to know about Radians vs Degrees?

Radians measure angle by arc length: one radian is the angle where the arc length equals the radius, so a full circle is 2π radians (≈6.283) — the same rotation as 360°. Converting: radians = degrees × (π/180), degrees = radians × (180/π). 90° = π/2 ≈ 1.571 rad; 45° = π/4 ≈ 0.785 rad. Degrees are intuitive for everyday use, but radians are essential in calculus because derivatives of trig functions (d/dx[sin(x)]=cos(x)) only hold true in this exact simple form when x is in radians — using degrees introduces an extra conversion constant everywhere. Most scientific and programming contexts (physics formulas, computer graphics, Python's math library) default to radians, so a common source of calculator and coding errors is forgetting to switch the mode, giving wildly wrong results like sin(90) in radian mode returning 0.894 instead of the expected 1.

What's the key thing to understand about the Ambiguous Case of the Sine Rule?

When using the Sine Rule to find an angle from two known sides and a non-included angle (the SSA case), there can be two valid triangles that fit the given information, not just one — this is called the ambiguous case. Because sine is positive in both the first and second quadrants, solving sin(B) = k for an unknown angle B gives two possible solutions, θ and 180°-θ, both of which can produce a geometrically valid triangle depending on the specific side lengths involved. To check which (or both) apply, you compare the given side lengths: if the side opposite the known angle is shorter than the other given side, both the acute and obtuse solutions might form a valid triangle (if their angle sum with the known angle stays under 180°), while a longer opposite side usually rules out the obtuse case entirely. Always testing both candidate angles against the triangle angle-sum rule is the only reliable way to catch this case.

Exact Trigonometric Values Table

These are the exact values every trigonometry student is expected to know without a calculator, derived from the 30-60-90 and 45-45-90 triangles:

Notice the symmetry: sin(θ) = cos(90° − θ) for every pair here (sin 30° = cos 60° = 0.5, sin 60° = cos 30° = √3/2 ≈ 0.866), which is the co-function identity. tan(90°) is undefined because cos(90°) = 0 would put a zero in the denominator — the tangent function has a vertical asymptote there.

Worked Example: Solving a Triangle with the Cosine and Sine Rules Together

A triangle has sides a = 8, b = 10, and the angle between them C = 40°. Find the third side c and the two remaining angles. Step 1 (Law of Cosines for the unknown side): c² = a² + b² − 2ab·cos(C) = 64 + 100 − 2(8)(10)cos(40°) = 164 − 160(0.76604) = 164 − 122.57 = 41.43, so c = √41.43 = 6.44. Step 2 (Law of Sines for angle A): sin(A)/a = sin(C)/c → sin(A) = a·sin(C)/c = 8 × 0.64279 / 6.44 = 5.1423/6.44 = 0.7985, so A = sin⁻¹(0.7985) = 53.0°. Since a < b, A must be acute — and the calculator correctly picks that branch rather than the obtuse alternative. Step 3 (angles sum to 180°): B = 180° − 40° − 53.0° = 87.0°. Check via the Law of Sines: sin(B)/b should equal sin(C)/c. sin(87.0°) = 0.99863, so sin(B)/b = 0.99863/10 = 0.099863, and sin(C)/c = 0.64279/6.44 = 0.099812 — matching to three decimal places (the small gap is rounding in the intermediate steps), confirming the triangle is solved correctly.

Key Trigonometric Identities Reference Table

Beyond the basic ratios, these identities are the backbone of simplifying trig expressions and solving trig equations:

As a quick check of the Pythagorean identity using the 30° exact values: sin(30°) = 0.5, cos(30°) = 0.8660. sin²(30°) + cos²(30°) = 0.25 + 0.75 = 1.00 ✓ — the identity holds for every angle, which is exactly why it's used so often to rewrite expressions involving sin into equivalent expressions involving cos, or to simplify equations before solving them. The double angle formula is also easy to verify numerically at 30°: sin(60°) should equal 2 sin(30°)cos(30°) = 2(0.5)(0.8660) = 0.8660, which indeed matches the exact value of sin(60°) = √3/2 = 0.8660 from the earlier reference table, confirming the identity holds in practice as well as in theory. These identities aren't just algebraic trivia — they're the main tool for simplifying expressions in calculus (integrating powers of sine and cosine, for instance) and for solving trigonometric equations that would otherwise have no clean closed-form solution without first rewriting them into an equivalent, simpler form. Engineering fields like signal processing and acoustics also lean heavily on these identities when decomposing complex waveforms into sums of simpler sine and cosine components.