Sine, Cosine, and Tangent as Ratios: SOH-CAH-TOA Without the Unit Circle

Trigonometric Functions as Side-Length Ratios (Right Triangles Only)

In this chapter, sine, cosine, and tangent are defined using only a right triangle and a chosen acute angle θ inside it. Each function is a ratio of two side lengths. That means the output of a trig function is a number (often a decimal or fraction), not an angle.

Because they are ratios, trig values are unitless: if you scale the triangle up or down, the side lengths change but the ratios stay the same.

Core Definitions (SOH-CAH-TOA)

For an acute angle θ in a right triangle:

• sin(θ) = opposite / hypotenuse
• cos(θ) = adjacent / hypotenuse
• tan(θ) = opposite / adjacent

A quick way to remember which sides go with which function is the mnemonic SOH-CAH-TOA: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent.

Important: The Ratio Depends on the Angle You Choose

The same right triangle contains two acute angles. If you switch from one acute angle to the other, what counts as “opposite” and “adjacent” swaps, so the trig ratios change. The hypotenuse stays the hypotenuse, but the other two sides trade roles depending on which angle is θ.

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Example Set A: Compute sin, cos, tan from a Given Triangle

Suppose a right triangle has these side lengths relative to angle θ:

• Opposite = 3
• Adjacent = 4
• Hypotenuse = 5

Compute each trig ratio by matching the function to the correct pair of sides.

Step-by-step: sin(θ)

1. Use the definition: sin(θ) = opposite / hypotenuse.
2. Substitute: sin(θ) = 3 / 5.
3. Optionally convert to decimal: sin(θ) = 0.6.

Step-by-step: cos(θ)

1. Use the definition: cos(θ) = adjacent / hypotenuse.
2. Substitute: cos(θ) = 4 / 5.
3. Optionally convert: cos(θ) = 0.8.

Step-by-step: tan(θ)

1. Use the definition: tan(θ) = opposite / adjacent.
2. Substitute: tan(θ) = 3 / 4.
3. Optionally convert: tan(θ) = 0.75.

Quick self-check (ratio reasonableness)

• sin(θ) and cos(θ) are each a leg divided by the hypotenuse, so each should be between 0 and 1 for an acute angle.
• tan(θ) is a leg divided by a leg, so it can be less than 1, equal to 1, or greater than 1 depending on the angle.

Example Set B: Reverse the Process (Interpret a Given Ratio)

Sometimes you are given a trig expression or value and asked what it means in terms of side lengths. The key is to translate the function back into “which side over which side.”

Example B1: Interpret sin(θ) = 3/5

Because sin(θ) means opposite/hypotenuse, the statement

sin(θ) = 3/5

means:

• The side opposite angle θ is in the ratio 3
• The hypotenuse is in the ratio 5

That does not force the triangle to have sides exactly 3 and 5; it means they are proportional. For instance, opposite/hypotenuse could be 6/10, 9/15, etc., and still equal 3/5.

Example B2: Interpret cos(θ) = 0.8

Because cos(θ) means adjacent/hypotenuse, the statement

cos(θ) = 0.8

means:

• adjacent / hypotenuse = 0.8

To see it as a fraction, write 0.8 = 8/10 = 4/5. So you can interpret it as:

• adjacent : hypotenuse = 4 : 5

Example B3: Interpret tan(θ) = 3/4

Because tan(θ) means opposite/adjacent, the statement

tan(θ) = 3/4

means:

• opposite : adjacent = 3 : 4

Again, many triangles can share this same tangent value as long as those two legs keep the same proportion (for example, 6:8, 9:12).

Mini-Checklist: Choosing sin, cos, or tan from the Sides Mentioned

When a problem mentions two sides (relative to θ), choose the trig function that uses exactly those two sides.

Checklist in action (fast selection)

• If you see the word hypotenuse, you are choosing between sine and cosine (since tangent does not use the hypotenuse).
• If you see both legs (opposite and adjacent), use tangent.
• If you are missing one of the needed sides for a ratio, you cannot compute that trig value yet (you would need more information).

Practice-Style Micro Examples

Micro Example 1: “Opposite is 10, hypotenuse is 26. Find sin(θ).”

1. Identify the two sides: opposite and hypotenuse.
2. Choose the function: sin(θ).
3. Compute: sin(θ) = 10/26 = 5/13.

Micro Example 2: “tan(θ)=2. What does that compare?”

1. Translate tangent: tan(θ) = opposite/adjacent.
2. So opposite/adjacent = 2, meaning opposite is twice adjacent.
3. Example ratio statement: opposite : adjacent = 2 : 1.

Micro Example 3: “Adjacent is 7, hypotenuse is 25. Find cos(θ).”

1. Identify the two sides: adjacent and hypotenuse.
2. Choose the function: cos(θ).
3. Compute: cos(θ) = 7/25.