Free Trigonometry Calculator with Steps
Calculate sin, cos, tan and all 6 trig functions step-by-step with exact values. Convert degrees to radians. Solve right triangles with SOH-CAH-TOA. Interactive unit circle and wave graphs. Perfect for GCSE and A-Level maths.
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Quick Examples
Trig Reference
Unit circle, identities & formulas
What is Trigonometry?
Trigonometry is the branch of mathematics that studies relationships between angles and sides of triangles. The word comes from Greek: "trigonon" (triangle) + "metron" (measure).
The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). In a right triangle, these ratios connect an angle to the lengths of the sides. Their reciprocals — cosecant (csc), secant (sec), and cotangent (cot) — complete the set of six trig functions.
The unit circle extends trigonometry beyond triangles: for any angle θ on a circle of radius 1, the point has coordinates (cos θ, sin θ). This is why sin and cos are always between −1 and 1, and why trig functions are periodic.
Real-world applications: navigation and GPS, sound waves and music, architecture, physics (projectile motion, forces), electrical engineering (AC circuits), computer graphics, and astronomy.
SOH-CAH-TOA & Trig Ratios
SOH
sin(θ) = Opposite / HypotenuseSine relates the side opposite the angle to the longest side (hypotenuse).
CAH
cos(θ) = Adjacent / HypotenuseCosine relates the side next to the angle to the hypotenuse.
TOA
tan(θ) = Opposite / AdjacentTangent relates the opposite side to the adjacent side. tan = sin/cos.
Reciprocal Functions
csc(θ)
1/sin(θ)
Hyp/Opp
sec(θ)
1/cos(θ)
Hyp/Adj
cot(θ)
1/tan(θ)
Adj/Opp
How to choose the right ratio: Label the sides as Opposite (across from the angle), Adjacent (next to the angle), and Hypotenuse (longest side). Then pick the function that connects your known value to your unknown.
Unit Circle & Special Angle Values
The unit circle is a circle of radius 1 centered at the origin. Every angle θ corresponds to a point (cos θ, sin θ) on the circle. This geometric definition extends trig functions to all angles, not just those in right triangles.
Special angles (0°, 30°, 45°, 60°, 90°) have exact trig values involving simple fractions and square roots. You must memorise these for GCSE and A-Level exams — calculators may not be allowed.
Degree ↔ Radian Conversion
Degrees → Radians: multiply by π/180
Radians → Degrees: multiply by 180/π
Quadrant rule (ASTC): "All Students Take Calculus" — Q1: All positive, Q2: Sin only, Q3: Tan only, Q4: Cos only. Use this to extend first-quadrant values to any angle.
Special Angle Exact Values
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undef |
| 120° | 2π/3 | √3/2 | -1/2 | -√3 |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 |
| 150° | 5π/6 | 1/2 | -√3/2 | -√3/3 |
| 180° | π | 0 | -1 | 0 |
Values for 180°–360° follow symmetry rules. Use the quadrant signs (ASTC) to determine the sign.
Trigonometric Identities Reference
Pythagorean Identities
sin²(θ) + cos²(θ) = 1
1 + tan²(θ) = sec²(θ)
1 + cot²(θ) = csc²(θ)
Derived from the Pythagorean theorem on the unit circle. The first is the most important.
Double Angle Formulas
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) - sin²(θ)
cos(2θ) = 2cos²(θ) - 1
cos(2θ) = 1 - 2sin²(θ)
tan(2θ) = 2tan(θ) / (1 - tan²(θ))
Sum & Difference
sin(A±B) = sinAcosB ± cosAsinB
cos(A±B) = cosAcosB ∓ sinAsinB
tan(A±B) = (tanA±tanB) / (1∓tanAtanB)
The \u2213 symbol means the sign is opposite to the \u00b1 above it.
Quotient & Reciprocal
tan(θ) = sin(θ) / cos(θ)
cot(θ) = cos(θ) / sin(θ)
csc(θ) = 1 / sin(θ)
sec(θ) = 1 / cos(θ)
cot(θ) = 1 / tan(θ)
Cofunction Identities
sin(θ) = cos(90° - θ)
cos(θ) = sin(90° - θ)
tan(θ) = cot(90° - θ)
Complementary angles sum to 90°. This is why "co-sine" means "complement of sine."
Half Angle Formulas
sin(θ/2) = ±√[(1-cosθ)/2]
cos(θ/2) = ±√[(1+cosθ)/2]
tan(θ/2) = sinθ / (1+cosθ)
The \u00b1 sign depends on the quadrant of θ/2.
Common Trigonometry Mistakes
Forgetting to check DEG/RAD mode
sin(90) = 0.8939... (in RAD mode)
sin(90°) = 1 (switch to DEG mode)
Always verify your calculator is in the correct angle mode. This is the #1 source of wrong answers.
Confusing inverse trig with reciprocal
sin⁻¹(x) = 1/sin(x)
sin⁻¹(x) = arcsin(x) ≠ 1/sin(x)
sin⁻¹ finds the angle, while 1/sin = csc is the reciprocal. Completely different operations!
Using SOH-CAH-TOA on wrong sides
Labelling sides without reference to the angle
Label O, A, H relative to YOUR specific angle
Opposite and Adjacent change depending on which angle you use. Hypotenuse is always the longest.
Forgetting the sign in other quadrants
sin(150°) = -√3/2
sin(150°) = +1/2 (Q2: sin is positive)
Use ASTC: All, Sin, Tan, Cos are positive in Q1, Q2, Q3, Q4 respectively.
Inputting degrees into a radians formula
Arc length = rθ with θ = 60 (degrees)
Convert first: θ = 60 × π/180 = π/3 rad
Formulas like arc length (s = rθ) and sector area (A = ½r²θ) always require radians.
Assuming tan is defined everywhere
tan(90°) = very large number
tan(90°) is undefined (÷ by zero)
tan = sin/cos, so it is undefined wherever cos = 0 (at 90°, 270°, etc.).
Worked Examples
Right triangle: angle = 35°, hypotenuse = 10 cm. Find opposite.
We know: angle = 35°, hypotenuse = 10
We want: opposite side
Choose SOH: sin(θ) = Opp / Hyp
sin(35°) = Opp / 10
Opp = 10 × sin(35°)
Opp = 10 × 0.5736 = 5.74 cm
Right triangle: opposite = 4, adjacent = 7. Find angle.
We know: opposite = 4, adjacent = 7
We want: the angle θ
Use TOA: tan(θ) = Opp / Adj = 4/7
θ = arctan(4/7)
θ = arctan(0.5714...)
θ = 29.7° (1 d.p.)
Find the exact value of sin(150°)
150° is in Quadrant 2 (between 90° and 180°)
Reference angle = 180° - 150° = 30°
sin(30°) = 1/2 (from special angles)
In Q2, sin is positive (ASTC: "Students")
sin(150°) = +1/2
Exact value: 1/2 = 0.5
Given sin(θ) = 3/5, find cos(θ) where 0° < θ < 90°
Use: sin²(θ) + cos²(θ) = 1
(3/5)² + cos²(θ) = 1
9/25 + cos²(θ) = 1
cos²(θ) = 1 - 9/25 = 16/25
cos(θ) = ±4/5
θ is in Q1, so cos(θ) = +4/5
Convert 225° to radians, then find cos(225°)
225° × (π/180) = 225π/180
= 5π/4 radians
225° is in Q3 (180° < 225° < 270°)
Reference angle = 225° - 180° = 45°
cos(45°) = √2/2
In Q3, cos is negative: cos(225°) = -√2/2
Given cos(θ) = 3/5 (Q1), find sin(2θ)
Use: sin(2θ) = 2sin(θ)cos(θ)
Find sin(θ): sin² + cos² = 1
sin² = 1 - 9/25 = 16/25, so sin = 4/5
sin(2θ) = 2 × (4/5) × (3/5)
sin(2θ) = 24/25
= 0.96
Exam Tips for Trigonometry
Always draw a diagram
For right triangle problems, sketch the triangle and label all known sides and angles. This makes it clear which trig ratio to use and prevents silly mistakes.
Check your calculator mode (DEG/RAD)
Before any trig calculation, verify your calculator is set to the correct angle unit. This is the single most common source of wrong answers in trig.
Memorise the special angle values
Learn sin, cos, tan for 0°, 30°, 45°, 60°, and 90°. Many exam questions specifically test these exact values without calculator access.
Use ASTC for signs in other quadrants
"All Students Take Calculus" tells you which functions are positive in Q1–Q4. Combine this with reference angles to find values in any quadrant.
Show your SOH-CAH-TOA choice
Write which ratio you are using and why. For example: "Using TOA because I know Opp and want Adj." This earns method marks even if you make an arithmetic slip.
Give exact values when asked
If the question says "exact value", give answers like √3/2 or 1/√2, not decimals. Rationalise the denominator if required: 1/√2 = √2/2.
Frequently Asked Questions
What is sin, cos, and tan?
These are the three primary trig functions. In a right triangle: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Remember the mnemonic SOH-CAH-TOA! They are the foundation of all trigonometry.
How do I convert degrees to radians?
Multiply degrees by π/180. For example: 90° × (π/180) = π/2 radians. Key values to memorise: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π.
What is the unit circle?
A circle with radius 1 centered at the origin. Any point at angle θ has coordinates (cosθ, sinθ). This is why sin and cos are always between -1 and 1. Our calculator includes an interactive unit circle visualisation.
What are inverse trig functions (arcsin, arccos, arctan)?
Inverse functions find the angle when you know the ratio. arcsin(0.5) = 30° because sin(30°) = 0.5. Important: arcsin and arccos only accept inputs from -1 to 1.
How do I solve a right triangle?
You need 2 pieces of information (one must be a side). Use SOH-CAH-TOA to find missing sides and inverse trig for missing angles. The Pythagorean theorem (a² + b² = c²) finds the third side.
When is tan undefined?
tan(θ) = sin(θ)/cos(θ), so it’s undefined when cos(θ) = 0. This happens at 90°, 270°, and all odd multiples of 90°. These appear as vertical asymptotes on the tan graph.
What is SOH-CAH-TOA and how do I use it?
SOH-CAH-TOA is a mnemonic: Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj. Label the triangle sides relative to your angle, then pick the ratio connecting your known and unknown values.
What are the Pythagorean trig identities?
The three identities are: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ. These are essential for simplifying expressions and proving identities at A-Level.
How do trig functions relate to the quadrants?
"All Students Take Calculus" (ASTC): Q1 = All positive, Q2 = Sin only, Q3 = Tan only, Q4 = Cos only. This determines the sign of any trig value for angles beyond 90°.
What are csc, sec, and cot?
The reciprocal functions: csc = 1/sin, sec = 1/cos, cot = 1/tan. They are undefined where their base function is zero. Important at A-Level and university level.
What are the special angle exact values I need to know?
For GCSE/A-Level, memorise: sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, and the corresponding cos values (reversed). tan(45°) = 1, tan(30°) = √3/3, tan(60°) = √3.
Is this suitable for GCSE and A-Level maths?
Yes! This calculator covers all GCSE topics (SOH-CAH-TOA, right triangles, special angles) and A-Level topics (unit circle, all 6 functions, inverse trig, identities, quadrant signs). The Learn Mode teaches exam-ready methods step-by-step.
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