Sin Cos Tan Calculator
Calculate all six trigonometric functions for any angle in degrees or radians.
📖 What are Trigonometric Functions?
Trigonometric functions describe the relationship between the angles and sides of a right triangle. They are among the most fundamental tools in mathematics, appearing in geometry, physics, engineering, music, computer graphics, and signal processing.
The three primary trigonometric functions are: - Sine (sin) - ratio of the opposite side to the hypotenuse - Cosine (cos) - ratio of the adjacent side to the hypotenuse - Tangent (tan) - ratio of the opposite side to the adjacent side (or sin/cos)
The three reciprocal functions are: - Cosecant (csc) = 1/sin - Secant (sec) = 1/cos - Cotangent (cot) = 1/tan
These six functions are defined for all angles (not just acute angles in right triangles) using the unit circle - a circle with radius 1 centred at the origin. On the unit circle, for any angle θ, the x-coordinate of the point on the circle equals cos(θ) and the y-coordinate equals sin(θ).
📐 Formula
sin(θ) = Opposite / Hypotenuse
cos(θ) = Adjacent / Hypotenuse
tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ)
csc(θ) = 1 / sin(θ)
sec(θ) = 1 / cos(θ)
cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ)
Degrees to Radians: rad = deg × π / 180
Radians to Degrees: deg = rad × 180 / π
📖 How to Use This Calculator
1
Enter an angle value in the input box.
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Select ° for degrees or rad for radians.
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Or click a common angle button (0°, 30°, 45°, 60°, 90°, 180°) for instant results.
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Click Calculate - all six trigonometric values are shown.
💡 Example Calculations
Example 1 - 30° angle
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sin(30°) = 0.5, cos(30°) = 0.866, tan(30°) = 0.5774
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csc(30°) = 2, sec(30°) = 1.1547, cot(30°) = 1.7321
Example 2 - Physics application
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A ramp is inclined at 35° to the horizontal. If the hypotenuse is 10 m:
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Height (opposite) = 10 × sin(35°) = 10 × 0.5736 = 5.736 m
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Base (adjacent) = 10 × cos(35°) = 10 × 0.8192 = 8.192 m
Frequently Asked Questions
What are sin, cos, and tan?
Sine (sin), cosine (cos), and tangent (tan) are the three primary trigonometric functions. For a right triangle with angle θ: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent (or sin/cos). They relate an angle to the ratios of sides in a right triangle.
What is the difference between degrees and radians?
Degrees and radians are two units for measuring angles. A full circle = 360° = 2π radians. To convert: radians = degrees × π/180. Degrees are more intuitive for everyday use; radians are preferred in calculus and physics because they simplify many formulas.
What are csc, sec, and cot?
These are the reciprocal trigonometric functions: csc(θ) = 1/sin(θ) (cosecant), sec(θ) = 1/cos(θ) (secant), cot(θ) = 1/tan(θ) (cotangent). They appear in more advanced trigonometry and calculus.
Why is tan(90°) undefined?
tan(θ) = sin(θ)/cos(θ). At 90°, cos(90°) = 0, which causes division by zero - hence tan(90°) is undefined. The tangent function approaches +∞ from the left and -∞ from the right at 90°.
What are the exact values of sin and cos for common angles?
sin(0°)=0, sin(30°)=0.5, sin(45°)=√2/2≈0.707, sin(60°)=√3/2≈0.866, sin(90°)=1. cos(0°)=1, cos(30°)=√3/2≈0.866, cos(45°)=√2/2≈0.707, cos(60°)=0.5, cos(90°)=0.
How do you remember the sin, cos, tan ratios?
The mnemonic SOH-CAH-TOA helps: SOH: Sin = Opposite / Hypotenuse. CAH: Cos = Adjacent / Hypotenuse. TOA: Tan = Opposite / Adjacent. In a right triangle, label the angle you are working with, then identify which side is opposite (facing the angle), adjacent (next to the angle, not the hypotenuse), and hypotenuse (longest side, opposite the right angle). Tan can also be remembered as sin divided by cos.
What are the exact values of sin, cos, tan for common angles?
Key exact values: sin(0) = 0, cos(0) = 1, tan(0) = 0. sin(30) = 1/2, cos(30) = sqrt(3)/2, tan(30) = 1/sqrt(3). sin(45) = sqrt(2)/2, cos(45) = sqrt(2)/2, tan(45) = 1. sin(60) = sqrt(3)/2, cos(60) = 1/2, tan(60) = sqrt(3). sin(90) = 1, cos(90) = 0, tan(90) = undefined. Memorising these 15 values covers the most common angles used in exams and engineering.
What are the reciprocal trigonometric functions?
The three reciprocal trig functions are: cosecant (csc) = 1/sin, secant (sec) = 1/cos, and cotangent (cot) = 1/tan. They appear in advanced calculus, physics, and engineering. For example, csc(30 degrees) = 1/sin(30 degrees) = 1/0.5 = 2. These functions are less common in basic trigonometry but essential in integration formulas (e.g. the integral of sec(x) is ln|sec(x) + tan(x)|) and in expressing identities. The Pythagorean identities also extend to these: 1 + tan^2(x) = sec^2(x) and 1 + cot^2(x) = csc^2(x).
What is the unit circle and why does it matter for trig functions?
The unit circle is a circle with radius 1 centered at the origin. Any point on it is (cos theta, sin theta) for angle theta. This gives the geometric meaning of sine and cosine as coordinates. It also explains why sin and cos are always between -1 and 1, why sin squared + cos squared = 1 (Pythagoras on the unit circle), and how trig functions extend beyond 90 degrees.