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Sine, Cosine, Tangent

The Hook

SOH CAH TOA.

What is it?

The three primary ratios of side lengths in a right-angled triangle.

Time Travel (History)

Ancient Indians and Arabs refined these ratios for astronomy and navigation long before calculators.

Cheat Code

SOH (Sin=Opp/Hyp), CAH (Cos=Adj/Hyp), TOA (Tan=Opp/Adj).

Practice This Now!

The Field Guide

In Plain English: Triangles have a secret code. If you know one angle, the ratio of the sides is locked in stone. 'Sine' is just a fancy word for how big the opposite side is compared to the slant (hypotenuse).

In The Real World: Carpentry. Calculating the exact angle to cut a rafter so the roof fits perfectly.

How To Do It

Identify your 'Target Angle' (theta).
Label the three sides: Hypotenuse (longest), Opposite (across from angle), Adjacent (touching angle).
Pick the function that matches the two sides you know or need (use SOH CAH TOA).
Write the equation and solve for x.

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Booby Trap!

Calculator mode! Using Degrees mode for Radian problems (or vice versa) will give you the wrong answer every time.

Real World Challenge

"You are placing a 12-foot ladder against a wall. Safety rules say the bottom angle must be 70 degrees. How high up the wall will the ladder reach?"

The Logic: We know Hypotenuse (12) and Angle (70). We want Opposite (height). SOH says Sin(70) = x/12. So, x = 12 * Sin(70). The ladder reaches approx 11.28 feet.

Training Drills

The Setup

/| H / | / | O /__ | A

H=Hypotenuse, O=Opposite, A=Adjacent

SOH Formula

Opposite Sin = ---------- Hypotenuse

Use when you don't care about the Adjacent side.