Trigonometric Ratios and Identities in Right Triangles
Trigonometric identities in right triangles are equations that hold for every acute angle, such as sin2A + cos2A = 1, tan A = sin A / cos A and sin A = cos(90° − A). In brief: if sin A = 3/5 in a right triangle, then cos A = 4/5 and tan A = 3/4, and all other ratios follow from one known ratio using these identities, without finding the angle.
The identities connect the six trigonometric ratios (sine, cosine, tangent, cotangent, secant and cosecant) and make it possible to simplify expressions, to prove other formulas and to find all ratios from one. They are derived from the Pythagorean theorem and the definitions of the ratios. This guide derives the basic identities, shows how to find all ratios from one, how to simplify and prove expressions and lists the identities in a table.
What are the basic identities?
For an acute angle A in a right triangle:
- Quotient identities: tan A = sin A/cos A and cot A = cos A/sin A.
- Reciprocal identities: cot A = 1/tan A, sec A = 1/cos A and csc A = 1/sin A.
- Pythagorean identity: sin2A + cos2A = 1.
- Cofunction identities: sin A = cos(90° − A), tan A = cot(90° − A).
Two further forms follow by division: 1 + tan2A = sec2A and 1 + cot2A = csc2A.
How are they derived?
Let the legs be a (opposite) and b (adjacent) and the hypotenuse c. Then sin A = a/c and cos A = b/c. The Pythagorean theorem a2 + b2 = c2 divided by c2 gives (a/c)2 + (b/c)2 = 1, that is sin2A + cos2A = 1. For the quotient: sin A/cos A = (a/c)/(b/c) = a/b = tan A. For the cofunction: the angle B = 90° − A has opposite side b and adjacent a, so sin B = b/c = cos A.
Dividing sin2A + cos2A = 1 by cos2A gives tan2A + 1 = 1/cos2A = sec2A.
How do you find all ratios from one?
- Draw a right triangle and label the known ratio with sides. For sin A = opposite/hypotenuse, give the opposite and the hypotenuse values.
- Find the third side with the Pythagorean theorem.
- Read off all other ratios.
Example 1: sin A = 5/13
Take opposite = 5 and hypotenuse = 13. The adjacent side is √(169 − 25) = 12. So cos A = 12/13, tan A = 5/12, cot A = 12/5, sec A = 13/12 and csc A = 13/5. Using the identity: cos A = √(1 − 25/169) = √(144/169) = 12/13 ✓ (positive for an acute angle).
Example 2: tan A = 2
Take opposite = 2 and adjacent = 1. The hypotenuse is √5. So sin A = 2/√5 = 2√5/5 ≈ 0.894, cos A = 1/√5 = √5/5 ≈ 0.447. Check: 0.8 + 0.2 = 1 ✓ (sin2A = 4/5 and cos2A = 1/5). The angle is about 63.43°.
Example 3: cos A = 0.6
sin A = √(1 − 0.36) = 0.8. tan A = 0.8/0.6 = 4/3 ≈ 1.333. The angle is 53.13°.
How do you simplify and prove expressions?
Rewrite everything in terms of sine and cosine, then simplify. To prove an identity, transform one side (usually the more complicated) until it equals the other side. Do not move terms across the equals sign in a proof.
Example 4: simplify
(1 − cos2A)/sin A = sin2A/sin A = sin A.
Example 5: prove tan A + cot A = 1/(sin A cos A)
Left side: sin A/cos A + cos A/sin A = (sin2A + cos2A)/(sin A cos A) = 1/(sin A cos A) ✓.
Example 6: prove (1 + tan2A) cos2A = 1
1 + tan2A = sec2A = 1/cos2A. So the product is 1 ✓.
Example 7: a numerical check
For A = 30°: sin = 0.5 and cos = 0.8660, so sin2 + cos2 = 0.25 + 0.75 = 1 ✓. And tan 30° = 0.5/0.866 = 0.5774 ✓.
Table of right-triangle identities
| Type | Identity | Check at A = 30° |
|---|---|---|
| Pythagorean | sin2A + cos2A = 1 | 0.25 + 0.75 = 1 |
| Quotient | tan A = sin A / cos A | 0.5/0.866 = 0.5774 |
| Reciprocal | cot A = 1 / tan A | 1/0.5774 = 1.7321 |
| Secant | sec A = 1 / cos A | 1/0.866 = 1.1547 |
| Cosecant | csc A = 1 / sin A | 1/0.5 = 2 |
| Cofunction | sin A = cos(90° − A) | sin 30° = cos 60° = 0.5 |
| Tangent form | 1 + tan2A = sec2A | 1 + 0.3333 = 1.3333 |
| Cotangent form | 1 + cot2A = csc2A | 1 + 3 = 4 |
Where do the identities come from?
Greek astronomers worked with relations between chords, equivalent to identities for sines. Ptolemy’s theorem on cyclic quadrilaterals gave formulas for the chord of a sum and difference. The modern notation and the systematic use of identities as algebraic tools appeared in the work of Leonhard Euler in the eighteenth century, who treated trigonometric functions as functions of a number instead of ratios of sides.
Where are they used today?
Identities simplify expressions in calculus (integrals such as ∫ sin2x dx rely on them), in physics (resolving vectors and wave interference) and in signal processing. They are also checked automatically by computer algebra software. In practical problems, the Pythagorean identity gives the missing component of a vector from the other one when the length is known.
Which mistakes are common?
Writing sin A2 for (sin A)2
sin2A means (sin A)2, not sin of A squared.
Cancelling “sin”
sin A/sin B is not A/B. A function name is not a factor.
Forgetting the sign when taking a root
For acute angles all ratios are positive. For other angles, the sign depends on the quadrant, as explained in the unit circle article.
Proving by working on both sides
In a proof, start from one side and transform it into the other, and do not assume the statement you want to prove.