Trig Functions Calculator

This trig functions calculator computes all six trigonometric functions — sine, cosine, tangent, cosecant, secant, and cotangent — for any angle you enter in degrees or radians. Alongside each result it plots the angle on a unit-circle diagram, finds the reference angle, and identifies the quadrant so you can see exactly why the value comes out positive or negative. Whether you are checking Precalculus homework or learning the unit circle for the first time, this trigonometry calculator turns a single angle into a complete, step-by-step picture in one click.

How to Use the Trig Functions Calculator

The tool is built to give a verifiable answer in seconds. You enter one angle and read back all six function values plus the supporting analysis. Follow these steps:

1. Enter your angle. Type the angle into the input field — any real number works, including negatives (such as −45) and angles larger than 360, like 750.
2. Choose degrees or radians. Select the unit that matches your problem. Degrees are typical in geometry and intro trig; radians are standard in calculus and physics.
3. Press Calculate. The calculator evaluates sin, cos, tan, csc, sec, and cot for that angle. Stuck? Use the Hint button to work through it yourself first.
4. Read the results. The answer panel lists all six values, the reference angle, and the quadrant. Open the Visual tab to see the angle plotted as a point on the unit circle, and the Steps tab for the full worked solution.

Because the diagram, reference angle, and quadrant come straight from the same calculation, you can cross-check your own homework against the picture instead of trusting a single number.

The Six Trigonometric Functions Explained

Trigonometry is built on six functions that relate an angle to ratios of side lengths. Three are primary — sine, cosine, and tangent — and the other three are their reciprocals. Knowing the primary three (and the mnemonic SOH-CAH-TOA) is enough to derive all six.

Sine, Cosine, and Tangent

For an acute angle in a right triangle, each function is a ratio of two sides relative to that angle. The mnemonic SOH-CAH-TOA captures all three: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. The opposite side faces the angle, the adjacent side touches it (and is not the hypotenuse), and the hypotenuse is the longest side, across from the right angle.

Tangent is also the ratio of the other two: tan θ = sin θ / cos θ. If you need to brush up on the sides of a right triangle themselves, the Pythagorean theorem calculator finds a missing side from the other two.

Cosecant, Secant, and Cotangent (the reciprocals)

The remaining three functions are simply the reciprocals of the primary three — flip the fraction. Cosecant pairs with sine, secant with cosine, and cotangent with tangent. Each is undefined wherever its partner equals zero, because you cannot divide by zero.

Note that cotangent is undefined where sine is zero, not where tangent is zero — because cot θ = cos θ / sin θ, the zero in the denominator is sin θ.

From Right Triangles to the Unit Circle

SOH-CAH-TOA only works for acute angles inside a right triangle — angles between 0° and 90°. To define the six functions for any angle, including 120°, 270°, or a negative angle, mathematicians extend the idea to the unit circle: a circle of radius 1 centered at the origin. This bridge is the single biggest comprehension step in trigonometry, and the diagram above the calculator shows it in action.

Picture an angle θ measured counterclockwise from the positive x-axis. The angle's terminal side crosses the unit circle at exactly one point. The coordinates of that point are defined to be:

So cosine is the x-coordinate and sine is the y-coordinate of the point on the circle. This matches SOH-CAH-TOA in the first quadrant: with a hypotenuse of 1, opposite/hypotenuse becomes just the height y, and adjacent/hypotenuse becomes just the horizontal distance x. The advantage is that x and y keep working past 90° — in the second quadrant x turns negative (so cosine is negative) while y stays positive (so sine stays positive). Tangent becomes y/x, the slope of the terminal side. When you press Calculate and open the Visual tab, the plotted point is precisely this (cos θ, sin θ). The circle calculator covers the broader geometry of circles, and once you move to oblique (non-right) triangles, the law of sines calculator takes over where SOH-CAH-TOA cannot.

Reference Angles and How to Find Them

A reference angle is the acute angle (between 0° and 90°) formed between the terminal side of your angle and the x-axis. It is the key shortcut for evaluating any angle by hand: the trig functions of an angle and its reference angle have the same absolute value — only the sign may differ, and the quadrant decides that sign. The calculator reports the reference angle for every input so you can see the connection.

To find the reference angle, first reduce the angle to within one full turn (0°–360°) by adding or subtracting 360° as needed, then apply the rule for its quadrant:

• Quadrant I (0°–90°): reference angle = θ
• Quadrant II (90°–180°): reference angle = 180° − θ
• Quadrant III (180°–270°): reference angle = θ − 180°
• Quadrant IV (270°–360°): reference angle = 360° − θ

For example, 210° lies in Quadrant III, so its reference angle is 210° − 180° = 30°. That tells you sin 210° has the same magnitude as sin 30° (= 0.5), and because Quadrant III sine is negative, sin 210° = −0.5.

Signs of Trig Functions by Quadrant (ASTC)

Once you know the reference angle's value, the quadrant tells you the sign. This is where the trig functions calculator's quadrant analysis pays off. The mnemonic "All Students Take Calculus" (ASTC) records which functions are positive in each quadrant, moving counterclockwise from Quadrant I: All, Sine, Tangent, Cosine.

A reciprocal always shares the sign of its partner (csc with sin, sec with cos, cot with tan), since flipping a fraction never changes its sign. Read the table together with the unit-circle diagram: in Quadrant II the point has negative x and positive y, which is exactly why cosine is negative and sine is positive there.

Degrees vs Radians

An angle can be measured in two units, and this calculator accepts both. Degrees split a full circle into 360 parts and are common in geometry and introductory trigonometry. Radians measure the angle by arc length on the unit circle, so a full circle is 2π radians (about 6.283), and they are the standard in calculus and physics because they make derivative formulas clean.

Convert between them with this rule:

For example, to convert 45° to radians, multiply: 45 × π/180 = π/4 ≈ 0.7854 radians. Useful landmarks: 180° = π, 90° = π/2, 60° = π/3, 45° = π/4, and 30° = π/6. If your homework gives an angle as a multiple of π, switch the calculator to radians; if it is a whole number like 60, use degrees.

Trig Function Values for Common Angles

A handful of "special" angles appear constantly, and their exact values are worth memorizing. This trig functions calculator returns these instantly, but the table below is a copy-ready reference for all six functions, with radian equivalents and undefined values flagged.

As decimals, √2/2 ≈ 0.7071, √3/2 ≈ 0.8660, and √3/3 ≈ 0.5774. Notice the symmetry: sin and cos swap values between 30° and 60°, and at 45° they are equal.

Why Some Functions Are Undefined

An "undefined" or error result is not a bug — it means a function would require dividing by zero. On the unit circle, certain functions have a zero in the denominator at specific angles:

• tan θ and sec θ are undefined at 90° and 270° (π/2 and 3π/2), where cos θ = 0 — both have cosine in the denominator.
• csc θ and cot θ are undefined at 0°, 180°, and 360° (0, π, 2π), where sin θ = 0 — both have sine in the denominator.

When you enter one of these angles, the calculator marks that specific function as undefined while still returning valid numbers for the others — for instance, at 90° it reports sin = 1 and cos = 0 but flags tan and sec as undefined.

Worked Examples

Here are two fully worked problems that mirror what the calculator produces, so you can follow the reasoning end to end.

Example 1 — sin, cos, and tan of 45°

1. Identify the quadrant and reference angle. 45° is in Quadrant I, so its reference angle is 45° and every function is positive (ASTC: "All").
2. Apply the definitions. sin 45° = √2/2 ≈ 0.7071, cos 45° = √2/2 ≈ 0.7071, and tan 45° = sin/cos = 1.
3. Confirm on the unit circle. The point is (0.7071, 0.7071) — equal x and y, which is why sine and cosine match. Result: sin 45° = 0.7071, Quadrant I, reference angle 45°.

Example 2 — all six functions of 210°

1. Find the reference angle. 210° is in Quadrant III, so reference angle = 210° − 180° = 30°.
2. Take the values of 30° and apply Quadrant III signs. In Quadrant III only tangent and cotangent are positive. So sin 210° = −1/2, cos 210° = −√3/2, tan 210° = +√3/3.
3. Take reciprocals for the last three. csc 210° = −2, sec 210° = −2√3/3, cot 210° = +√3. Each reciprocal keeps its partner's sign.

Real-World Uses of Trigonometric Functions

Trigonometric functions are far more than a classroom exercise — they describe anything that involves angles, rotation, or repeating waves. A few everyday applications:

• Navigation and surveying: bearings and triangulation use sine and cosine to find distances and positions that cannot be measured directly.
• Physics: breaking a force or velocity into horizontal and vertical components relies on cos θ and sin θ.
• Engineering and construction: the tangent of an angle gives roof pitch, ramp slope, and the rise-over-run of a staircase.
• Astronomy: the angular position of stars and planets is computed with trig functions.
• Sound and signals: audio, light, and AC electricity are modeled as sine waves, so these functions underpin music and electronics.

When a problem involves a triangle without a right angle, the law of cosines calculator handles cases where you know two sides and the included angle.

Common Mistakes

Studying for AP Precalculus? Trig functions anchor Unit 3 (Trigonometric and Polar Functions) — 30–35% of the exam. See which trig identities are actually tested or the 2027 score calculator.

Frequently Asked Questions

• What are the six trig functions?

  The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. Sine, cosine, and tangent are the primary functions; cosecant, secant, and cotangent are their reciprocals (1/sin, 1/cos, and 1/tan respectively). This calculator returns all six for any angle at once.

• How do I find sin, cos, and tan of an angle?

  Enter the angle into the trig functions calculator, choose degrees or radians, and press Calculate. By hand, find the reference angle, look up its sine, cosine, and tangent, then apply the quadrant sign using ASTC. The tool shows each of these steps in the Steps tab.

• What is a reference angle?

  A reference angle is the acute angle (0° to 90°) between the terminal side of your angle and the x-axis. Trig functions of an angle and its reference angle share the same absolute value, so it lets you evaluate any angle from the special-angle values, with the quadrant supplying the sign.

• How do I switch between degrees and radians?

  Use the degrees/radians toggle next to the angle input. To convert manually, multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. For example, 45° equals π/4, which is about 0.7854 radians.

• Why is tan(90°) undefined?

  Tangent equals sin θ divided by cos θ, and cos 90° = 0. Dividing by zero has no defined result, so tan 90° is undefined rather than a number. The same applies to secant at 90°, and to cosecant and cotangent at 0° and 180°, where sine is zero.

• What are csc, sec, and cot?

  They are the reciprocal functions: cosecant (csc) = 1/sin, secant (sec) = 1/cos, and cotangent (cot) = 1/tan. Each is undefined wherever its partner equals zero. Note that cosecant pairs with sine and secant pairs with cosine, which students often reverse.

• What is the sign of each trig function in each quadrant?

  Use "All Students Take Calculus" (ASTC): all functions are positive in Quadrant I, only sine and cosecant in Quadrant II, only tangent and cotangent in Quadrant III, and only cosine and secant in Quadrant IV. The calculator labels the quadrant so you can apply this automatically.

• What are the trig values of 30°, 45°, and 60°?

  sin 30° = 1/2, cos 30° = √3/2; sin 45° = cos 45° = √2/2; sin 60° = √3/2, cos 60° = 1/2. The tangents are √3/3, 1, and √3 respectively. As decimals, √2/2 ≈ 0.7071 and √3/2 ≈ 0.8660. See the common-angles table above for all six functions.

• Can this calculator do inverse trig functions (arcsin, arccos, arctan)?

  This tool evaluates the six forward functions — it takes an angle and returns sin, cos, tan, csc, sec, and cot. Inverse functions work the other way, taking a ratio and returning an angle, so they are outside this calculator's scope. Enter an angle, not a ratio, to use it.

• Is this trig functions calculator free?

  Yes. Every calculator on MathInSite is completely free with no login, no sign-up, and no paywall on the steps. You get all six function values, the unit-circle diagram, the reference angle, the quadrant, and the full worked solution at no cost. Browse all our free math calculators from the home page.

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