Trigonometry Calculator
Compute all six trig functions for any angle in degrees or radians.
Trig results depend on degrees vs radians.
sin(30 °)
0.5
sin(θ) = opposite ÷ hypotenuse
Formula used
sin(θ) = opposite ÷ hypotenuse
sin(30°)
Angle in degrees
30°
Angle in radians
0.5235987756 rad
sin
0.5
cos
0.8660254038
tan
0.5773502692
csc
2
sec
1.1547005384
cot
1.7320508076
Quadrant note
The angle is in Quadrant I.
Special angle
Special angle: 30° = π/6 radians.
Trigonometry Formulas and Identities
Sine
sin(θ) = opposite ÷ hypotenuse
Cosine
cos(θ) = adjacent ÷ hypotenuse
Tangent
tan(θ) = opposite ÷ adjacent
Reciprocal Functions
csc(θ) = 1 ÷ sin(θ), sec(θ) = 1 ÷ cos(θ), cot(θ) = 1 ÷ tan(θ)
Pythagorean Identity
sin²(θ) + cos²(θ) = 1
Degree/Radian Conversion
radians = degrees × π ÷ 180; degrees = radians × 180 ÷ π
Inverse Trig
arcsin(x), arccos(x), and arctan(x) find angles from ratios.
Variable Explanations
θ
Angle.
opposite
Side across from the angle.
adjacent
Side next to the angle.
hypotenuse
Longest side of a right triangle.
radians
Angle measure based on arc length.
degrees
Angle measure where a full circle is 360°.
inverse trig
Finding an angle from a trig ratio.
undefined
Result cannot be calculated because division by zero occurs.
Right-Triangle Diagram and Unit-Circle Meaning
Sine uses opposite and hypotenuse, cosine uses adjacent and hypotenuse, and tangent uses opposite and adjacent. On the unit circle, cosine is the x-coordinate and sine is the y-coordinate.
Worked Examples
sin(30°)
Formula or rule: sin(θ) = opposite ÷ hypotenuse
Substitution: sin(30°)
Answer: 0.5
A 30-60-90 triangle has sine 1/2.
cos(60°)
Formula or rule: cos(θ) = adjacent ÷ hypotenuse
Substitution: cos(60°)
Answer: 0.5
Cosine compares adjacent side to hypotenuse.
tan(45°)
Formula or rule: tan(θ) = opposite ÷ adjacent
Substitution: tan(45°)
Answer: 1
In a 45-45-90 triangle, opposite and adjacent are equal.
180° to radians
Formula or rule: radians = degrees × π ÷ 180
Substitution: 180 × π ÷ 180
Answer: π radians
A straight angle is π radians.
π/2 radians to degrees
Formula or rule: degrees = radians × 180 ÷ π
Substitution: π/2 × 180 ÷ π
Answer: 90°
π/2 radians equals a right angle.
arcsin(0.5)
Formula or rule: arcsin(x) gives the angle whose sine is x
Substitution: arcsin(0.5)
Answer: 30° or π/6
Inverse sine returns an angle.
Right triangle ratio
Formula or rule: sin(θ) = opposite ÷ hypotenuse
Substitution: 3 ÷ 5
Answer: 0.6
Opposite 3 and hypotenuse 5 gives sine 0.6.
tan(90°)
Formula or rule: tan(θ) = sin(θ) ÷ cos(θ)
Substitution: 1 ÷ 0
Answer: undefined
Tangent is undefined when cosine is zero.
Special angle
Formula or rule: cos(45°)
Substitution: √2 ÷ 2
Answer: ≈ 0.7071
Special angles often have exact forms.
Degree/radian mistake
Formula or rule: sin(90°) vs sin(90 radians)
Substitution: different angle modes
Answer: different results
Always check the active angle mode.
Degrees vs Radians
Degrees
Degrees divide a full circle into 360 parts. Geometry and many everyday angle problems often use degrees.
Radians
Radians measure angles using radius and arc length. 180° equals π radians and 90° equals π/2 radians.
Sine, Cosine, Tangent, and Reciprocal Functions
Sine
Opposite ÷ hypotenuse.
Cosine
Adjacent ÷ hypotenuse.
Tangent
Opposite ÷ adjacent.
Cosecant
Reciprocal of sine.
Secant
Reciprocal of cosine.
Cotangent
Reciprocal of tangent.
Undefined values
Reciprocal functions are undefined when the denominator is zero.
Common Use Cases
Common Trigonometry Mistakes
Understanding Your Result
Sine result
Opposite-to-hypotenuse ratio.
Cosine result
Adjacent-to-hypotenuse ratio.
Tangent result
Opposite-to-adjacent ratio.
Inverse trig result
Angle that produces the given ratio.
Degree/radian conversion
Same angle expressed in another unit.
Undefined result
Function is not defined for that input.
Exact value
Symbolic result for special angles.
Frequently Asked Questions
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