Step by Step
SOH
Sine = Opposite / Hypotenuse
For a given angle in a right triangle, sine equals the length of the side opposite that angle divided by the hypotenuse (the longest side, always opposite the right angle).
CAH
Cosine = Adjacent / Hypotenuse
Cosine equals the length of the side adjacent to the angle (the non-hypotenuse side touching the angle) divided by the hypotenuse.
TOA
Tangent = Opposite / Adjacent
Tangent equals the length of the opposite side divided by the adjacent side — notice the hypotenuse doesn't appear in this ratio at all.
Key
Identifying opposite vs. adjacent correctly
"Opposite" and "adjacent" are always defined relative to the specific angle you're working with — if you look at a different angle in the same triangle, which side counts as opposite and which counts as adjacent will switch.
Applied Walkthrough
1
A right triangle has a hypotenuse of 10, and the angle θ has an opposite side of 6 and an adjacent side of 8.
2
Using SOH: sin(θ) = opposite/hypotenuse = 6/10 = 0.6.
3
Using CAH: cos(θ) = adjacent/hypotenuse = 8/10 = 0.8.
4
Using TOA: tan(θ) = opposite/adjacent = 6/8 = 0.75. Notice tan(θ) can also be found by dividing sin(θ) by cos(θ): 0.6/0.8 = 0.75 — confirming the relationship between all three ratios.
Exam Application
Exams test whether you can correctly identify which side is opposite versus adjacent for a specific angle, and whether you can set up the correct ratio (SOH, CAH, or TOA) based on which sides are given or needed.
⚠ Common Trap
The most common trap is confusing which side is "opposite" and which is "adjacent" when the triangle is drawn in an unfamiliar orientation — always identify these relative to the specific angle you're using, not by the triangle's visual position on the page.
✓ Quick Self-Check
1. What does SOH stand for?
Sine = Opposite / Hypotenuse.
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2. What does CAH stand for?
Cosine = Adjacent / Hypotenuse.
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3. What does TOA stand for?
Tangent = Opposite / Adjacent.
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4. Which side is always the longest and opposite the right angle?
The hypotenuse.
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5. How can you find tan(θ) if you already know sin(θ) and cos(θ)?
tan(θ) = sin(θ) / cos(θ).
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