Sin Cos Tan Calculator

Q: Is sin⁻¹ the same as 1/sin?

No. They are two different things. sin⁻¹(x) means arcsine. It takes a ratio and gives you back an angle. Example: sin⁻¹(0.5) = 30°. 1/sin(θ) is the cosecant, written csc(θ). Example: 1/sin(30°) = 2. The small −1 next to sin is not an exponent. It is a symbol that means "inverse function." Mixing these up is one of the most common trig mistakes.

Q: Why does sin(30) give −0.988 instead of 0.5?

Because the angle was read as 30 radians , not 30 degrees. 30 radians is about 1719°, which lands in a totally different spot on the circle. That is why the answer is negative. Fix it by setting the angle unit to degrees, or by converting first: 30° = 30 × π/180 ≈ 0.5236 radians. Then sin(0.5236) = 0.5.

Q: Which trig functions are positive in each quadrant?

Use the phrase All Students Take Calculus to remember it: Quadrant I (0° to 90°): All are positive. Quadrant II (90° to 180°): only Sine (and csc) is positive. Quadrant III (180° to 270°): only Tangent (and cot) is positive. Quadrant IV (270° to 360°): only Cosine (and sec) is positive. This works because cosine is the x-value on the unit circle and sine is the y-value. Wherever x or y is negative, that function is negative too.

Q: When should you use sin, cos, or tan to solve a triangle?

Pick the one that uses the two sides you care about: Use sin when you have the opposite side and the hypotenuse. Use cos when you have the adjacent side and the hypotenuse. Use tan when you have the opposite and adjacent sides (no hypotenuse). Label the sides first: the hypotenuse is the longest side, opposite is across from your angle, adjacent is beside it. Then choose the ratio that matches the two sides you know.

Q: Can sine or cosine be greater than 1?

No. Sine and cosine always stay between −1 and 1. On the unit circle, the radius is 1. Sine is the height of the point and cosine is the sideways distance. Neither can be longer than the radius. In a right triangle, sine and cosine are a leg divided by the hypotenuse, and the hypotenuse is always the longest side. So the answer is never more than 1. Tangent, secant, cosecant, and cotangent are different. They can be any size, even in the millions.

Q: What is the exact value of sin(45°)?

sin(45°) = √2 ⁄ 2 ≈ 0.7071. Cos(45°) is the same number. This comes from a 45-45-90 triangle. If both legs are 1, the hypotenuse is √2. So sine = 1/√2, which is the same as √2/2. Since sine and cosine match, tan(45°) = 1.

Q: Why does arcsin only give angles between −90° and 90°?

Because many angles share the same sine. Both 30° and 150° have a sine of 0.5. A function can only return one answer, so math rules pick one main range. arcsin: −90° to 90° arccos: 0° to 180° arctan: −90° to 90° If your real problem has an angle outside that range, find the second answer yourself. For sine, use 180° − your answer. You can also add 360° to get coterminal angles.

Q: What does sin²θ + cos²θ equal?

It always equals 1 , for every angle. This is called the Pythagorean identity. It comes from the unit circle, where the point (cos θ, sin θ) sits on a circle of radius 1, so x² + y² = 1. Try it with 30°: 0.5² + 0.8660² = 0.25 + 0.75 = 1. It is handy when you know one ratio and need the other.

Q: How many degrees is 1 radian?

One radian is about 57.2958° . A full circle is 2π radians, which is 360°. Divide 360 by 2π and you get 57.2958. Quick swaps: Radians to degrees: multiply by 180/π Degrees to radians: multiply by π/180 π radians = 180°, π/2 radians = 90°

Q: Is sin(−θ) the same as sin(θ)?

No. sin(−θ) = −sin(θ). Flipping the angle flips the sign. Cosine works differently: cos(−θ) = cos(θ). The sign stays the same. Tangent acts like sine: tan(−θ) = −tan(θ). On the unit circle the reason is easy to see. A negative angle turns the other way, so the height (sine) flips below the axis, but the sideways distance (cosine) does not change.

Sin Cos Tan Calculator

Angle (Degrees)	Angle (Radians)	Load
0°	0
30°	π/6
45°	π/4
60°	π/3
90°	π/2
120°	2π/3
135°	3π/4
150°	5π/6
180°	π
270°	3π/2
360°	2π

Angle (Degrees)

Angle (Radians)

Load

0°

30°

π/6

45°

π/4

60°

π/3

90°

π/2

120°

2π/3

135°

3π/4

150°

5π/6

180°

270°

3π/2

360°

2π

Introduction

This Sin Cos Tan Calculator gives the value of any trig function. Type an angle, pick degrees, radians, or gradians, and read the answer. The inverse functions (arcsin, arccos, and arctan) work the other way and find the angle from a ratio.

The calculator handles six functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). For every angle you enter, it shows all six values at once, so you can compare them in one look.

You also get a step-by-step solution that shows the math, a unit circle picture with the (cos θ, sin θ) point, and a graph of the curve with your answer marked. If a value is undefined, like tan(90°), the tool tells you why.

Click any row in the common angles table to load angles like 30°, 45°, 60°, or 90° in one tap. Or load a worked example, such as a roof pitch or a ladder angle, to see trigonometry used in real life. It is a simple way to check homework, study for a test, or solve a triangle at work.

How to use our Sin Cos Tan Calculator

Pick a trig function, type your angle or ratio, and choose your units. The calculator shows the answer, all six trig values for that angle, a unit circle, a function graph, and step-by-step math.

Select Function: Tap the function you want. Use sin, cos, tan, csc, sec, or cot to turn an angle into a number. Use arcsin, arccos, or arctan to turn a ratio back into an angle.

Enter Angle (θ) or Ratio (x): Type your number here. For sin, cos, and tan, type the angle. For arcsin and arccos, type a ratio between −1 and 1. For arctan, any number works. The × button clears the box.

Angle Unit (input): Choose degrees, radians, or gradians to match the angle you typed. This box shows up only for the direct functions.

Answer Angle Unit (output): Choose the unit for your answer angle. This box shows up only for arcsin, arccos, and arctan. The answer is also listed in all three units.

Decimal Precision: Pick how many decimal places you want, from 2 up to 8, or choose full precision for every digit.

Calculate and Reset: Press Calculate to see your result. Press Reset to go back to the starting values.

Common Angles table: Click any row, like 30°, 45°, or 90°, to load that angle right into the calculator.

Worked Examples: Click Load Example to fill in a real-life problem, such as roof pitch or ladder angle, and see how it is solved.

Sin, Cos, and Tan Explained

Sine (sin), cosine (cos), and tangent (tan) are the three main trigonometric functions. They connect the angles of a right triangle to the lengths of its sides. If you know an angle and one side, these functions help you find the other sides. If you know two sides, they help you find the angle.

The Three Basic Ratios

In a right triangle, the longest side is the hypotenuse. The side across from your angle is the opposite side. The side next to your angle is the adjacent side.

sin(θ) = opposite ÷ hypotenuse
cos(θ) = adjacent ÷ hypotenuse
tan(θ) = opposite ÷ adjacent

A common way to remember this is SOH-CAH-TOA.

The Reciprocal Functions

Three more functions are just the flipped versions of the first three:

csc(θ) = 1 ÷ sin(θ) (cosecant)
sec(θ) = 1 ÷ cos(θ) (secant)
cot(θ) = cos(θ) ÷ sin(θ) (cotangent)

When the bottom of one of these fractions equals zero, the answer is undefined. For example, tan(90°) is undefined because cos(90°) = 0, and you cannot divide by zero.

Inverse Functions: Finding the Angle

Sometimes you already know the ratio and want the angle. That is what arcsin, arccos, and arctan do. They work backwards. For example, arctan(0.5) ≈ 26.57°, which is the angle of a roof that rises 1 foot for every 2 feet across.

Arcsin and arccos only accept numbers from −1 to 1, because sine and cosine can never be bigger than 1 or smaller than −1. Arctan accepts any number.

Degrees, Radians, and Gradians

Angles can be measured three ways. A full circle is 360 degrees, 2π radians (about 6.2832), or 400 gradians. Degrees are used in everyday work and school. Radians are used in higher math, physics, and computer code. Gradians show up in surveying and some engineering jobs.

Degrees to radians: multiply by π/180
Radians to degrees: multiply by 180/π
Gradians to radians: multiply by π/200

The Unit Circle

The unit circle is a circle with a radius of 1 centered at (0, 0). For any angle, the point on the circle has coordinates (cos θ, sin θ). This is why sine and cosine always stay between −1 and 1, and it explains why the values repeat every 360°. Angles that land on the same spot are called coterminal angles, like 30° and 390°.

Common Angle Values

These angles show up often and are worth memorizing:

sin(0°) = 0, cos(0°) = 1, tan(0°) = 0
sin(30°) = 0.5, cos(30°) ≈ 0.8660, tan(30°) ≈ 0.5774
sin(45°) ≈ 0.7071, cos(45°) ≈ 0.7071, tan(45°) = 1
sin(60°) ≈ 0.8660, cos(60°) = 0.5, tan(60°) ≈ 1.7321
sin(90°) = 1, cos(90°) = 0, tan(90°) is undefined

Where You Use Trigonometry

Builders use tangent to set roof pitch and ramp slope. Physics students use sine and cosine to split a force or a thrown ball's speed into side-to-side and up-and-down parts. Sailors and pilots use them for bearings and distance. Surveyors measure land heights with them. Sound waves, light waves, and electric current are all drawn as sine curves.

Reading the Graphs

Sine and cosine make smooth waves that repeat every 360° and never go past 1 or −1. Cosine is just the sine wave shifted 90° to the left. Tangent looks different: it climbs from negative infinity to positive infinity and repeats every 180°, with vertical breaks called asymptotes at 90°, 270°, and so on. Those breaks are exactly where tangent is undefined.

Formulas used

Degrees to radians conversion

\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}

Gradians to radians conversion

\theta_{\text{rad}} = \theta_{\text{grad}} \times \frac{\pi}{200}

Radians to degrees and gradians

\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}, \qquad \theta_{\text{grad}} = \theta_{\text{rad}} \times \frac{200}{\pi}

Tangent from sine and cosine

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, \quad \cos(\theta) \ne 0

Reciprocal functions (cosecant, secant, cotangent)

\csc(\theta) = \frac{1}{\sin(\theta)}, \qquad \sec(\theta) = \frac{1}{\cos(\theta)}, \qquad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}

Inverse trigonometric functions (angle from ratio)

\theta = \arcsin(x),\; \theta = \arccos(x) \;\text{ for } -1 \le x \le 1; \qquad \theta = \arctan(x) \;\text{ for all real } x

Unit circle point coordinates

(x,\, y) = (\cos\theta,\, \sin\theta)

Coterminal angle used for the diagram

\theta_{c} = \theta \bmod 2\pi, \quad 0 \le \theta_{c} < 2\pi

Frequently asked questions

Is sin⁻¹ the same as 1/sin?

No. They are two different things.

sin⁻¹(x) means arcsine. It takes a ratio and gives you back an angle. Example: sin⁻¹(0.5) = 30°.
1/sin(θ) is the cosecant, written csc(θ). Example: 1/sin(30°) = 2.

The small −1 next to sin is not an exponent. It is a symbol that means "inverse function." Mixing these up is one of the most common trig mistakes.

Why does sin(30) give −0.988 instead of 0.5?

Because the angle was read as 30 radians, not 30 degrees.

30 radians is about 1719°, which lands in a totally different spot on the circle. That is why the answer is negative.

Fix it by setting the angle unit to degrees, or by converting first: 30° = 30 × π/180 ≈ 0.5236 radians. Then sin(0.5236) = 0.5.

Which trig functions are positive in each quadrant?

Use the phrase All Students Take Calculus to remember it:

Quadrant I (0° to 90°): All are positive.
Quadrant II (90° to 180°): only Sine (and csc) is positive.
Quadrant III (180° to 270°): only Tangent (and cot) is positive.
Quadrant IV (270° to 360°): only Cosine (and sec) is positive.

This works because cosine is the x-value on the unit circle and sine is the y-value. Wherever x or y is negative, that function is negative too.

When should you use sin, cos, or tan to solve a triangle?

Pick the one that uses the two sides you care about:

Use sin when you have the opposite side and the hypotenuse.
Use cos when you have the adjacent side and the hypotenuse.
Use tan when you have the opposite and adjacent sides (no hypotenuse).

Label the sides first: the hypotenuse is the longest side, opposite is across from your angle, adjacent is beside it. Then choose the ratio that matches the two sides you know.

Can sine or cosine be greater than 1?

No. Sine and cosine always stay between −1 and 1.

On the unit circle, the radius is 1. Sine is the height of the point and cosine is the sideways distance. Neither can be longer than the radius.

In a right triangle, sine and cosine are a leg divided by the hypotenuse, and the hypotenuse is always the longest side. So the answer is never more than 1.

Tangent, secant, cosecant, and cotangent are different. They can be any size, even in the millions.

What is the exact value of sin(45°)?

sin(45°) = √2 ⁄ 2 ≈ 0.7071. Cos(45°) is the same number.

This comes from a 45-45-90 triangle. If both legs are 1, the hypotenuse is √2. So sine = 1/√2, which is the same as √2/2.

Since sine and cosine match, tan(45°) = 1.

Why does arcsin only give angles between −90° and 90°?

Because many angles share the same sine. Both 30° and 150° have a sine of 0.5. A function can only return one answer, so math rules pick one main range.

arcsin: −90° to 90°
arccos: 0° to 180°
arctan: −90° to 90°

If your real problem has an angle outside that range, find the second answer yourself. For sine, use 180° − your answer. You can also add 360° to get coterminal angles.

What does sin²θ + cos²θ equal?

It always equals 1, for every angle.

This is called the Pythagorean identity. It comes from the unit circle, where the point (cos θ, sin θ) sits on a circle of radius 1, so x² + y² = 1.

Try it with 30°: 0.5² + 0.8660² = 0.25 + 0.75 = 1. It is handy when you know one ratio and need the other.

How many degrees is 1 radian?

One radian is about 57.2958°.

A full circle is 2π radians, which is 360°. Divide 360 by 2π and you get 57.2958.

Quick swaps:

Radians to degrees: multiply by 180/π
Degrees to radians: multiply by π/180
π radians = 180°, π/2 radians = 90°

Is sin(−θ) the same as sin(θ)?

No. sin(−θ) = −sin(θ). Flipping the angle flips the sign.

Cosine works differently: cos(−θ) = cos(θ). The sign stays the same.

Tangent acts like sine: tan(−θ) = −tan(θ).

On the unit circle the reason is easy to see. A negative angle turns the other way, so the height (sine) flips below the axis, but the sideways distance (cosine) does not change.

How can you find sin, cos, and tan without a calculator?

For the common angles, use this pattern for sine: take √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, and 90°.

That gives sine values of 0, 0.5, 0.7071, 0.8660, and 1. For cosine, read the same list backwards. For tangent, divide sine by cosine.

For other angles you need a calculator or a trig table, since the math behind them uses long infinite sums.

How do you turn a roof pitch into an angle?

Use arctan. Divide the rise by the run, then take the inverse tangent.

Example: a 4/12 pitch rises 4 inches for every 12 inches across. 4 ÷ 12 = 0.3333, and arctan(0.3333) ≈ 18.43°.

A 6/12 pitch gives about 26.57°, and a 12/12 pitch gives exactly 45°. The same math works for ramps, stairs, and driveway slopes.

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