Trigonometric Identities Calculator: Check Any Identity for an Angle

The six trigonometric functions of an angle, and every standard identity worked out on both sides so you can see that they agree.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter an angle A (and a second angle B for the sum, difference and product formulas). The calculator gives sine, cosine and tangent, the three reciprocals (cosecant, secant, cotangent), the quadrant the angle lies in, and its reference angle.

It then works out each standard identity on the left side and on the right side separately and shows both numbers. The Pythagorean identities, the co-function identities, the double angle and half angle formulas, the sum and difference formulas and the product-to-sum formulas should all agree. It is a quick way to check you have remembered a formula correctly, or to see how it behaves at a particular angle.

Formulas used

  • \sin^2 A + \cos^2 A = 1
  • \sin 2A = 2\sin A\cos A
  • \sin(A \pm B) = \sin A\cos B \pm \cos A\sin B
  • \sin A\cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]

Worked example

With the default numbers in the calculator above:

  1. sin A: 0.5
  2. cos A: 0.86603
  3. tan A: 0.57735
  4. cot A: 1.7321
  5. sec A: 1.1547
  6. cosec A: 2
  7. Where the angle lies: Quadrant I (all of sin, cos and tan are positive); reference angle 30°
  8. Pythagorean: sin²A + cos²A: 1 (both sides agree)
  9. Pythagorean: 1 + tan²A: 1.3333 (both sides agree)
  10. Pythagorean: 1 + cot²A: 4 (both sides agree)
  11. Co-function: sin(90° − A): 0.86603 (both sides agree)
  12. Co-function: cos(90° − A): 0.5 (both sides agree)

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

Why does a function say undefined?

Tangent and secant are undefined where cosine is zero (90 and 270 degrees). Cotangent and cosecant are undefined where sine is zero (0 and 180 degrees). Any identity that needs one of them says so for that angle.

Degrees or radians?

Choose the unit. 180 degrees is pi radians, so 90 degrees is about 1.5708 radians and 45 degrees is about 0.7854.

What are the three forms of cos 2A?

cos squared minus sin squared, 2 cos squared minus 1, and 1 minus 2 sin squared. They are equal; each is shown.

What is a reference angle?

The acute angle between the terminal side of the angle and the x axis. The size of every function is its value at the reference angle, and the quadrant gives its sign.

Related calculators

Further reading: Standard engineering mathematics texts (trigonometric identities).

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