Trigonometric ratios appear from Class 10 through engineering and physics. Enter an angle in degrees or radians to get all six ratios, with undefined values such as tan 90° clearly marked.
Enter the angle.
Choose degrees or radians.
Read all six ratios.
tan: tan θ = sin θ ÷ cos θ
Reciprocals: cosec θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = 1 ÷ tan θ
Degrees to radians: rad = degrees × π ÷ 180
Enter an angle in degrees or radians, and this calculator gives all six trigonometric ratios together: sine, cosine, tangent, cosecant, secant and cotangent. Where a ratio does not exist, such as tan 90° or cot 0°, it says not defined instead of printing a huge meaningless number.
Trigonometry begins in Class 10 with right triangles and heights and distances, and grows in Class 11 into functions of any angle. It is essential in physics for resolving forces and describing waves, in surveying for heights and distances, in engineering for alternating current and in computer graphics for rotation. Most students memorise the standard table for 0°, 30°, 45°, 60° and 90°; the calculator covers every other angle and confirms the standard values in decimal form.
1. If the angle is in degrees, convert it to radians with θ × π ÷ 180. Radians are used by most computing tools.
2. In a right triangle, sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent.
3. For any angle, place it on the unit circle: cos θ is the x-coordinate and sin θ is the y-coordinate of the point reached.
4. Compute tan θ = sin θ ÷ cos θ, which is undefined when cos θ = 0.
5. Take reciprocals: cosec θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = cos θ ÷ sin θ.
6. Use the quadrant (All, Sin, Tan, Cos) to check the signs of your answers.
In a right triangle the ratios depend only on the angle, not on the triangle's size, because all right triangles with the same acute angle are similar. That definition only covers angles between 0° and 90°. To extend it, draw a circle of radius 1 centred at the origin and turn a radius through angle θ from the positive x-axis. The point it reaches is (cos θ, sin θ). For acute angles this matches the triangle definition, and it works for any angle, including negative ones and angles beyond 360°.
Since the point (cos θ, sin θ) lies on a circle of radius 1, Pythagoras gives sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ. The signs follow from the coordinates: in the second quadrant x is negative, so cos is negative while sin stays positive. The memory aid All Silver Tea Cups lists which ratios are positive in quadrants one to four.
At 90° the point on the unit circle is (0, 1), so cos 90° = 0 and tan 90° = 1 ÷ 0 is undefined, as is sec 90°. At 0° and 180°, sin is 0, so cot and cosec are undefined. Because π cannot be stored exactly, a computer finds cos 90° as about 6 × 10⁻¹⁷ rather than 0. The calculator rounds anything smaller than 10⁻¹² to zero so these cases are flagged correctly. Exact surd values such as √3/2 are shown as decimals.
Ishita is solving a heights-and-distances problem where the angle of elevation of a temple tower's top is 60°, and she wants all the ratios for that angle.
Angle in radians: 60 × π ÷ 180 = 1.047198 rad
Primary ratios: sin = 0.866025, cos = 0.5, tan = sin ÷ cos = 1.732051
Reciprocal ratios: cosec = 1 ÷ sin = 1.154701, sec = 1 ÷ cos = 2, cot = cos ÷ sin = 0.57735
Answer: sin 0.866025; cos 0.5; tan 1.732051
Entering a degree value while the unit is set to radians, or the other way round, which gives completely different answers.
Confusing cosec with the inverse function sin⁻¹. cosec θ is 1 ÷ sin θ, not the angle whose sine is θ.
Writing sin²θ as sin(θ²) instead of (sin θ)².
Forgetting that ratios can be negative for angles beyond 90°.
Swapping the opposite and adjacent sides, which swaps sin with cos and tan with cot.
Solving Class 10 heights and distances problems for towers, trees and buildings.
Resolving forces and velocities into components in physics.
Finding slope angles for ramps, roofs and roads in civil work.
Describing waves, pendulums and alternating current signals.
Rotating shapes and camera views in graphics and game programming.
Why is tan 90° undefined?
Because cos 90° = 0, and tan = sin ÷ cos would require division by zero.
Does the calculator give exact surd values?
It gives decimal values. For standard angles, sin 30° = 0.5 exactly and sin 45° = 1/√2 ≈ 0.7071.