How to Use SOH CAH TOA in Right Triangles?
SOH CAH TOA gives the three basic trigonometric ratios in a right-angled triangle: sine, cosine and tangent. Relative to an angle θ, sin θ equals opposite divided by hypotenuse, cos θ equals adjacent divided by hypotenuse, and tan θ equals opposite divided by adjacent. These ratios help find unknown sides or angles.
What Is SOH CAH TOA?
The hypotenuse is the side opposite the 90° angle and is always the longest side. The opposite and adjacent sides are named with respect to the chosen angle θ. Thus, the same side can be opposite for one angle and adjacent for another angle.
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Trigonometric Ratios: SOH-CAH-TOA
Learn the three basic trigonometric ratios—sine, cosine and tangent—using the SOH-CAH-TOA mnemonic and a simple memory trick for right-angled triangles.
SOH CAH TOA Formula & Tricks
Important Formulas
Use sine when the opposite side and hypotenuse are involved.
Use cosine when the adjacent side and hypotenuse are involved.
Use tangent when the opposite and adjacent sides are involved.
Choose the inverse ratio matching the two known sides.
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Quick Tricks
Use the mnemonic SOH CAH TOA: Sine uses Opposite and Hypotenuse, Cosine uses Adjacent and Hypotenuse, and Tangent uses Opposite and Adjacent.
First mark the right angle and the given angle. Then identify the two sides involved. Do not name opposite or adjacent without referring to the chosen angle.
SOH CAH TOA Concepts
Identifying the Three Sides
The hypotenuse is never one of the two sides forming the chosen angle θ. The adjacent side is the non-hypotenuse side that touches θ. The opposite side does not touch θ. For a triangle with sides 5, 12 and 13, the side 13 is the hypotenuse when the angle opposite it is 90°.
Using SOH CAH TOA for a Missing Side
For example, if θ = 30° and the hypotenuse is 10, then O = H sin θ = 10 × 1/2 = 5. If θ = 60° and the adjacent side is 8, then O = A tan θ = 8√3.
Finding an Angle with Inverse Ratios
For opposite = 3 and hypotenuse = 5, sin θ = 3/5, so θ = sin⁻¹(3/5) ≈ 36.87°. The calculator must be in degree mode when the answer is required in degrees.
Related Ratios in a Right Triangle
These reciprocals are not part of the SOH CAH TOA mnemonic but follow directly from the three basic ratios. Also, tan θ = sin θ/cos θ whenever cos θ is non-zero.
SOH CAH TOA Visual Guide
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Right triangle for SOH CAH TOA
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SOH CAH TOA Revision Points
Core ratios and rules for right-triangle trigonometry.
- sin θ = O/H, cos θ = A/H and tan θ = O/A.
- The hypotenuse is opposite the 90° angle and is the longest side.
- Opposite and adjacent sides are identified relative to the chosen angle θ.
- Use inverse trigonometric functions to find an unknown angle.
- Use degree mode for answers expressed in degrees.
- In every right triangle, H² = O² + A².
SOH CAH TOA FAQs
What is the SOH CAH TOA formula?
SOH CAH TOA means sin θ = O/H, cos θ = A/H and tan θ = O/A, where O is opposite, A is adjacent and H is hypotenuse.
Which side is the hypotenuse?
The hypotenuse is the side opposite the 90° angle. It is the longest side of a right-angled triangle.
What is sin 30° using SOH CAH TOA?
sin 30° = 1/2. Therefore, in a right triangle, the opposite side is half the hypotenuse when the reference angle is 30°.
How do you find a missing side when the angle and hypotenuse are known?
Use sine: O = H sin θ if the opposite side is required, or A = H cos θ if the adjacent side is required. For H = 10 and θ = 30°, O = 10 × 1/2 = 5.
How do you find an angle when the opposite and adjacent sides are known?
Use θ = tan⁻¹(O/A). For O = 5 and A = 12, θ = tan⁻¹(5/12) ≈ 22.62°.
What is the difference between adjacent and opposite sides?
The opposite side lies across from the selected angle θ. The adjacent side touches θ but is not the hypotenuse; both names change if the reference angle changes.