SOH CAH TOA is a memory aid for the three main trig ratios of a right triangle: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.

How do you find sin, cos and tan of a right triangle?

Label the sides relative to the angle theta, then divide. For a 3-4-5 triangle and the angle opposite the side 3: sin = 3/5 = 0.6, cos = 4/5 = 0.8 and tan = 3/4 = 0.75.

What are the reciprocal trig functions?

Cosecant is 1 / sin = Hypotenuse / Opposite, secant is 1 / cos = Hypotenuse / Adjacent, and cotangent is 1 / tan = Adjacent / Opposite.

What are the exact trig values for 30, 45 and 60 degrees?

sin 30 = 1/2, cos 30 = sqrt(3)/2, tan 30 = sqrt(3)/3 (about 0.5774); sin 45 = cos 45 = sqrt(2)/2 (about 0.7071) and tan 45 = 1; sin 60 = sqrt(3)/2 (about 0.8660), cos 60 = 1/2 and tan 60 = sqrt(3) (about 1.7321).

Which side is opposite, adjacent and the hypotenuse?

The hypotenuse is the longest side, across from the right angle. The opposite side is across from the angle theta, and the adjacent side is the leg that touches theta (next to it, but not the hypotenuse).

Why is tan 90 degrees undefined?

At 90 degrees the adjacent side shrinks to 0, and tan = Opposite / Adjacent would divide by zero. For the same reason sec 90, csc 0 and cot 0 are undefined.

SOH CAH TOA Formula Sheet: Trig Ratios Table

Right Triangle Trigonometric Ratios (SOH CAH TOA) Quick Sheet

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SOH CAH TOA gives the three main trig ratios of a right triangle: sine, cosine and tangent. This sheet lists all six ratios with their reciprocals, exact values for key angles, and a live calculator.

SOH: Sinesin(θ) = O / HOpposite over Hypotenuse

CAH: Cosinecos(θ) = A / HAdjacent over Hypotenuse

TOA: Tangenttan(θ) = O / AOpposite over Adjacent

Cosecantcsc(θ) = H / OReciprocal of sine

Secantsec(θ) = H / AReciprocal of cosine

Cotangentcot(θ) = A / OReciprocal of tangent

Trig Ratio Calculator

Function

Angle θ (degrees, 0 to 90)

Tip: open this page with ?fn=sin (or cos, tan, csc, sec, cot) to start with that function and highlight its row. Add &angle=45 to set the angle too.

SOH CAH TOA and Reciprocal Trig Ratios
Trig Function	Abbreviation	Ratio Formula	Opposite/Adjacent Relation	Reciprocal
Sine	sin(θ)	Opposite / Hypotenuse	SOH	Cosecant (csc)
Cosine	cos(θ)	Adjacent / Hypotenuse	CAH	Secant (sec)
Tangent	tan(θ)	Opposite / Adjacent	TOA	Cotangent (cot)
Cosecant	csc(θ)	Hypotenuse / Opposite	1 / sin(θ)	Sine (sin)
Secant	sec(θ)	Hypotenuse / Adjacent	1 / cos(θ)	Cosine (cos)
Cotangent	cot(θ)	Adjacent / Opposite	1 / tan(θ)	Tangent (tan)

Exact Trig Values for 0°, 30°, 45°, 60° and 90°
Angle θ	sin	cos	tan	csc	sec	cot
0°	0	1	0	undefined	1	undefined
30°	1/2	√3/2	√3/3	2	2√3/3	√3
45°	√2/2	√2/2	1	√2	√2	1
60°	√3/2	1/2	√3	2√3/3	2	√3/3
90°	1	0	undefined	1	undefined	0

Trig Table: 0° to 90° in 5° Steps (4 decimal places)
Angle θ	sin	cos	tan	csc	sec	cot
0°	0	1	0	undefined	1	undefined
5°	0.0872	0.9962	0.0875	11.4737	1.0038	11.4301
10°	0.1736	0.9848	0.1763	5.7588	1.0154	5.6713
15°	0.2588	0.9659	0.2679	3.8637	1.0353	3.7321
20°	0.342	0.9397	0.364	2.9238	1.0642	2.7475
25°	0.4226	0.9063	0.4663	2.3662	1.1034	2.1445
30°	0.5	0.866	0.5774	2	1.1547	1.7321
35°	0.5736	0.8192	0.7002	1.7434	1.2208	1.4281
40°	0.6428	0.766	0.8391	1.5557	1.3054	1.1918
45°	0.7071	0.7071	1	1.4142	1.4142	1
50°	0.766	0.6428	1.1918	1.3054	1.5557	0.8391
55°	0.8192	0.5736	1.4281	1.2208	1.7434	0.7002
60°	0.866	0.5	1.7321	1.1547	2	0.5774
65°	0.9063	0.4226	2.1445	1.1034	2.3662	0.4663
70°	0.9397	0.342	2.7475	1.0642	2.9238	0.364
75°	0.9659	0.2588	3.7321	1.0353	3.8637	0.2679
80°	0.9848	0.1736	5.6713	1.0154	5.7588	0.1763
85°	0.9962	0.0872	11.4301	1.0038	11.4737	0.0875
90°	1	0	undefined	1	undefined	0

SOH CAH TOA Solved on 6 Classic Right Triangles
Sides (O, A, H)	sinθ = O / H	cosθ = A / H	tanθ = O / A	Angle θ
(3, 4, 5)	3/5 = 0.6	4/5 = 0.8	3/4 = 0.75	36.8699°
(5, 12, 13)	5/13 = 0.3846	12/13 = 0.9231	5/12 = 0.4167	22.6199°
(8, 15, 17)	8/17 = 0.4706	15/17 = 0.8824	8/15 = 0.5333	28.0725°
(7, 24, 25)	7/25 = 0.28	24/25 = 0.96	7/24 = 0.2917	16.2602°
(20, 21, 29)	20/29 = 0.6897	21/29 = 0.7241	20/21 = 0.9524	43.6028°
(9, 40, 41)	9/41 = 0.2195	40/41 = 0.9756	9/40 = 0.225	12.6804°

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Frequently Asked Questions

Question	Answer
What is SOH CAH TOA?	SOH CAH TOA is a memory aid for the three main trig ratios of a right triangle: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.
How do you find sin, cos and tan of a right triangle?	Label the sides relative to the angle θ, then divide. For a 3-4-5 triangle and the angle opposite the side 3: sin = 3/5 = 0.6, cos = 4/5 = 0.8 and tan = 3/4 = 0.75.
What are the reciprocal trig functions?	Cosecant is 1 / sin = Hypotenuse / Opposite, secant is 1 / cos = Hypotenuse / Adjacent, and cotangent is 1 / tan = Adjacent / Opposite.
What are the exact trig values for 30, 45 and 60 degrees?	sin 30 = 1/2, cos 30 = √3/2, tan 30 = √3/3 (about 0.5774); sin 45 = cos 45 = √2/2 (about 0.7071) and tan 45 = 1; sin 60 = √3/2 (about 0.8660), cos 60 = 1/2 and tan 60 = √3 (about 1.7321).
Which side is opposite, adjacent and the hypotenuse?	The hypotenuse is the longest side, across from the right angle. The opposite side is across from the angle θ, and the adjacent side is the leg that touches θ (next to it, but not the hypotenuse).
Why is tan 90 degrees undefined?	At 90 degrees the adjacent side shrinks to 0, and tan = Opposite / Adjacent would divide by zero. For the same reason sec 90, csc 0 and cot 0 are undefined.

Right Triangle Trigonometric Ratios (SOH CAH TOA) Quick Sheet

Trig Ratio Calculator

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Frequently Asked Questions

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