Sin Cos Tan Calculator

Exact Values Table — GCSE and A-Level

UK exam boards (Edexcel, AQA, OCR) require students to know these exact trig values without a calculator. The table below covers all angles tested at GCSE and A-Level.

Angle (°)	Radians	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
120°	2π/3	√3/2	−1/2	−√3
135°	3π/4	√2/2	−√2/2	−1
150°	5π/6	1/2	−√3/2	−1/√3
180°	π	0	−1	0
270°	3π/2	−1	0	undefined
360°	2π	0	1	0

What Are Sin, Cos and Tan?

Trigonometry is the branch of mathematics concerned with the relationships between angles and side lengths in triangles. The three fundamental trigonometric functions — sine (sin), cosine (cos) and tangent (tan) — form the backbone of this subject and are essential at both GCSE and A-Level in England, Scotland, Wales and Northern Ireland.

In a right-angled triangle with an angle θ, the three sides are labelled relative to that angle:

• Hypotenuse: the longest side, opposite the right angle (90°)
• Opposite: the side directly opposite the angle θ
• Adjacent: the side next to angle θ (not the hypotenuse)

The mnemonic SOH-CAH-TOA is universally used in UK schools and is one of the most remembered pieces of mathematical notation. It allows students to quickly recall which sides relate to which function.

The Unit Circle — Beyond Right-Angled Triangles

While the SOH-CAH-TOA definition works perfectly for right-angled triangles, it breaks down for angles greater than 90° or negative angles. The unit circle provides a more powerful definition that works for all angles.

The unit circle is a circle of radius 1 centred at the origin. For any angle θ measured anticlockwise from the positive x-axis:

• cos(θ) = the x-coordinate of the point on the unit circle
• sin(θ) = the y-coordinate of the point on the unit circle
• tan(θ) = sin(θ) / cos(θ) = the slope of the radius line

This extends trig to all real numbers and is the foundation of A-Level and university mathematics. It explains why, for example, sin(150°) = sin(30°) = 0.5, because the points are symmetric about the y-axis.

Degrees vs Radians

Two systems exist for measuring angles: degrees and radians.

• Degrees divide a full circle into 360 equal parts. This is the system used throughout GCSE.
• Radians measure angles by the arc length on the unit circle. A full circle = 2π radians ≈ 6.2832 radians.

Conversion formulas:

• Degrees to radians: multiply by π/180    (e.g., 90° × π/180 = π/2)
• Radians to degrees: multiply by 180/π    (e.g., π/3 × 180/π = 60°)

At A-Level, particularly when differentiating or integrating trig functions, radians are mandatory. The derivative of sin(x) is cos(x) only when x is in radians.

Inverse Trigonometric Functions

The inverse functions — arcsin (sin¹), arccos (cos¹), and arctan (tan¹) — take a ratio and return the corresponding angle. They are essential when you know the sides of a triangle and need to find an angle.

• arcsin(x): returns angle in range −90° to 90° (or −π/2 to π/2 rad)
• arccos(x): returns angle in range 0° to 180° (or 0 to π rad)
• arctan(x): returns angle in range −90° to 90° (or −π/2 to π/2 rad)

Important: sin¹(x) in this context means the inverse sine function, NOT 1/sin(x). These are entirely different expressions. 1/sin(x) is cosecant (cosec or csc).

The Sine Rule

The sine rule applies to any triangle (not just right-angled). For a triangle with sides a, b, c and opposite angles A, B, C:

Worked example: In triangle ABC, a = 8 cm, A = 35°, B = 60°. Find side b.
b/sin(60°) = 8/sin(35°)
b = 8 × sin(60°) / sin(35°) = 8 × 0.8660 / 0.5736 ≈ 12.08 cm

The Cosine Rule

The cosine rule is used when two sides and the included angle are known, or when all three sides are known and an angle is required.

Worked example: Triangle with b = 5, c = 7, A = 40°. Find a.
a² = 25 + 49 − 2(5)(7)cos(40°) = 74 − 70 × 0.7660 = 74 − 53.62 = 20.38
a = √20.38 ≈ 4.51

Pythagorean Identities

Trig identities are equations that are true for all valid angle values. The most important at A-Level:

• sin²(θ) + cos²(θ) = 1 — the fundamental Pythagorean identity
• 1 + tan²(θ) = sec²(θ) — dividing the identity by cos²(θ)
• 1 + cot²(θ) = cosec²(θ) — dividing the identity by sin²(θ)

These identities are used to simplify expressions, solve trig equations, and prove further identities. The formula sin²(θ) + cos²(θ) = 1 is given on Edexcel and AQA formula sheets, but understanding how to derive it from the unit circle is valuable for deeper exam questions.

CAST Diagram and Quadrants

The CAST diagram helps students remember which trig functions are positive in each quadrant of the unit circle:

• Q1 (0°–90°): All (sin, cos, tan) are positive
• Q2 (90°–180°): Sine only is positive
• Q3 (180°–270°): Tangent only is positive
• Q4 (270°–360°): Cosine only is positive

CAST stands for Cosine, All, Sine, Tangent reading anticlockwise from Q4. This is essential for solving trig equations where multiple solutions exist within a given range (e.g., 0° ≤ θ ≤ 360°).

Real-World Applications of Trigonometry in the UK

Trigonometry is far from abstract — it underpins countless practical applications encountered in UK careers and everyday life:

• Architecture and construction: Calculating roof pitches, ramp gradients, and structural load angles on UK building projects.
• Navigation: Bearings in geography lessons use compass directions combined with sine and cosine to find distances and positions.
• Engineering: Civil, mechanical and electrical engineers routinely apply trig to resolve forces, design components, and analyse waveforms.
• Physics: Projectile motion, wave equations, and alternating current (AC) circuits all rely on sin and cos functions.
• Surveying: Land surveyors in the UK use trigonometric levelling to measure elevation differences across terrain.
• Computer graphics: Rotation of objects in 2D and 3D games and simulations is computed using sin and cos.

Common Mistakes to Avoid

• Wrong calculator mode: Always verify your calculator is in DEG mode for degree problems. Casio fx-83/85/991 series show “D” at the top of the screen for degrees.
• Confusing inverse notation: sin¹(x) means arcsin(x), not 1/sin(x). The latter is cosecant.
• Forgetting multiple solutions: Trig equations typically have multiple solutions. Always check the full range stated in the question (e.g., −360° to 360°) and use the CAST diagram to find all answers.
• Rounding too early: Keep full precision in intermediate steps and only round the final answer. Premature rounding causes significant errors in multi-step problems.
• Applying SOH-CAH-TOA to non-right-angled triangles: SOH-CAH-TOA only applies when there is a right angle present. For other triangles, use the sine rule or cosine rule.

Frequently Asked Questions

What are sin, cos and tan?

Sin (sine), cos (cosine) and tan (tangent) are the three primary trigonometric ratios. In a right-angled triangle: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. They extend to all angles via the unit circle definition.

Should I use degrees or radians?

Use degrees for GCSE. Use radians for A-Level calculus (differentiation and integration of trig functions require radians). Always check your calculator mode before computing — a wrong mode is the most common source of trig errors on UK exams.

What are the exact trig values I must know?

Edexcel, AQA and OCR require exact values at 0°, 30°, 45°, 60° and 90°. See the table above for all values. These are non-negotiable memorisation tasks at GCSE and A-Level.

What is the difference between sin and arcsin?

sin(θ) takes an angle and returns a ratio. arcsin(x) (also written sin¹(x)) takes a ratio and returns an angle. They are inverse operations. Example: sin(30°) = 0.5, so arcsin(0.5) = 30°.

How do I solve a trig equation like sin(x) = 0.5 for 0° ≤ x ≤ 360°?

Step 1: Find the principal value: arcsin(0.5) = 30°. Step 2: Use the CAST diagram. Since sin is positive in Q1 and Q2, the second solution is 180° − 30° = 150°. So x = 30° or x = 150°.

Why is tan(90°) undefined?

Because tan(θ) = sin(θ)/cos(θ), and cos(90°) = 0. Division by zero is undefined. On a graph, tan(x) has vertical asymptotes at x = 90°, 270° and all odd multiples of 90°.

What is the Pythagorean identity?

sin²(θ) + cos²(θ) = 1 for all values of θ. This follows from Pythagoras’ theorem applied to the unit circle. It is the most important trig identity and the basis for deriving others.