Trigonometry Calculator — sin, cos, tan & Right Triangles

A trigonometry calculator evaluates sin, cos, tan, csc, sec and cot of an angle and solves right triangles. For 30°, sin 30° = 1/2, cos 30° = √3/2 and tan 30° = √3/3 ≈ 0.5774. In a right triangle with legs 3 and 4, the hypotenuse is √(3² + 4²) = 5, so sin A = 3/5 = 0.6 and A ≈ 36.87°.

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How to use the trigonometry calculator

1. Trig functions: choose sin, cos, tan, csc, sec or cot, type the angle and pick degrees or radians. You get the exact value for multiples of 15° (π/12), the decimal, the quadrant, the reference angle, all six functions and a unit circle.
2. Inverse functions: choose sin⁻¹, cos⁻¹, tan⁻¹, csc⁻¹, sec⁻¹ or cot⁻¹ and type a value such as 0.5, −√2/2 or 3/4. You get the principal angle in degrees and radians and every solution between 0° and 360°.
3. Right triangle: enter any two values, with at least one side: two sides, or one side and one acute angle. The calculator finds the missing sides and angles, the area, the perimeter and all six ratios.
4. Identities: type an angle to see 22 standard identities with your angle substituted and both sides compared.
5. Open Show steps for the full working, and use Copy result to paste the answer into your notes.

SOH CAH TOA and the six trig functions

In a right triangle, label the sides from the point of view of the angle θ: the opposite side is across from θ, the adjacent side touches θ, and the hypotenuse is the longest side, across from the right angle. SOH CAH TOA gives the three main ratios, and the other three are their reciprocals.

Right triangles only cover angles between 0° and 90°. For any other angle, use the unit circle: the angle ends at the point (cos θ, sin θ). The unit circle calculator shows that point for any angle.

Exact values of special angles

For angles beyond 90°, find the reference angle (the acute angle to the x-axis) and attach the sign of the quadrant: all functions are positive in Quadrant I, only sine in II, only tangent in III and only cosine in IV (“All Students Take Calculus”). The reference angle calculator covers this step in detail.

Worked examples

sin 150° and sec(5π/6)

150° is in Quadrant II with reference angle 180° − 150° = 30°. Sine is positive there, so sin 150° = sin 30° = 1/2. 5π/6 is the same angle in radians. Cosine is negative in Quadrant II, so cos(5π/6) = −√3/2 and sec(5π/6) = 1/(−√3/2) = −2/√3 = −2√3/3 ≈ −1.1547.

Why tan 90° is undefined

tan θ = sin θ / cos θ. At 90° the unit circle point is (0, 1), so tan 90° = 1/0, which has no value. The same happens to sec θ at 90° and 270°, and to csc θ and cot θ at 0° and 180°.

Find a missing angle: opposite 5, hypotenuse 10

sin θ = opposite / hypotenuse = 5/10 = 0.5, so θ = sin⁻¹(0.5) = 30°. The third side is √(10² − 5²) = √75 = 5√3 ≈ 8.6603, and the other acute angle is 90° − 30° = 60°.

Find missing sides: hypotenuse 8, angle 30°

The opposite side is 8 × sin 30° = 8 × 1/2 = 4, and the adjacent side is 8 × cos 30° = 8 × √3/2 = 4√3 ≈ 6.9282. The area is 4 × 4√3 / 2 = 8√3 ≈ 13.8564.

Inverse trig functions and their ranges

An inverse trig function gives back an angle. Because many angles share the same sine, each inverse returns one angle from a fixed principal range:

For example, sin⁻¹(0.5) = 30°, but sin 150° = 0.5 as well, so 150° is a second solution between 0° and 360°. The calculator lists both. For all six inverse functions in depth, use the inverse trig calculator.

Key trigonometric identities

• Pythagorean: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ.
• Quotient: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ.
• Even-odd: sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ.
• Cofunction: sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.
• Double angle: sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1.
• Half angle: sin(θ/2) = ±√((1 − cos θ)/2) and cos(θ/2) = ±√((1 + cos θ)/2), with the sign set by the quadrant of θ/2.

Check one with θ = 30°: sin 60° = √3/2, and 2 sin 30° cos 30° = 2 × 1/2 × √3/2 = √3/2.

Common mistakes in trigonometry

• Wrong angle mode. sin 30 in radian mode means the sine of 30 radians, about −0.9880, not 0.5. Match the calculator mode to your angle.
• Mixing up opposite and adjacent. They depend on which angle you are using. The hypotenuse never changes.
• Reading sin⁻¹ x as 1/sin x. sin⁻¹ is the inverse function (arcsin). The reciprocal 1/sin x is csc x.
• Forgetting the sign. cos 120° is −1/2, not 1/2, because cosine is negative in Quadrant II.
• Rounding too early. Keep exact values or several decimals until the last step, especially when you reuse a side to find another.

Trigonometry practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you find a missing angle in trigonometry?

Pick the ratio that uses the two sides you know, then apply its inverse. With opposite 5 and hypotenuse 10, sin θ = 5/10 = 0.5, so θ = sin⁻¹(0.5) = 30°. With opposite 3 and adjacent 4, θ = tan⁻¹(3/4) ≈ 36.87°.

What is the trigonometry formula for right triangles?

SOH CAH TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent, together with the Pythagorean theorem a² + b² = c² and the fact that the two acute angles add to 90°.

Can you do trigonometry without a calculator?

Yes, for special angles. Learn sin 30° = 1/2, sin 45° = √2/2 and sin 60° = √3/2 (the cosines are the same values in reverse order), then use reference angles and the ASTC signs. sin 30° = 1/2, cos 45° = √2/2 and tan 60° = √3 all come from the 30-60-90 and 45-45-90 triangles.

How do you use a scientific calculator for trigonometry?

Set the mode to DEG or RAD to match your angle, press sin, cos or tan, then type the angle. For csc, sec and cot, take the reciprocal: csc 30° = 1 ÷ sin 30° = 2. For an angle, use SHIFT or 2nd with sin, cos or tan to get sin⁻¹, cos⁻¹ or tan⁻¹.

What is cot in trigonometry?

Cotangent is the reciprocal of tangent: cot θ = 1/tan θ = adjacent/opposite = cos θ / sin θ. For example, cot 45° = 1 and cot 30° = √3 ≈ 1.732. It is undefined where sin θ = 0, such as 0° and 180°.

Is trigonometry calculus?

No. Trigonometry studies angles, triangles and the trig functions, and it is usually taught in geometry, Algebra 2 and precalculus. Calculus uses trig functions heavily, for example the derivative of sin x is cos x, so a solid trig base makes calculus much easier.

Related calculators

• Unit circle calculator
• Inverse trig calculator
• Reference angle calculator
• Law of sines and cosines calculator (triangles without a right angle)
• Special right triangles calculator
• Degrees to radians calculator