Right Triangle Trigonometry Formula

The Formula

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}

When to use: Imagine a ramp leaning against a wall. The steepness depends on the ratio of how high the wall is to how long the ramp is. Trigonometry gives names to these ratios: sine is how high compared to the ramp, cosine is how far along the ground compared to the ramp, and tangent is how high compared to how far along the ground. No matter how big or small the ramp, if the angle is the same, these ratios stay the same.

Quick Example

In a right triangle with angle \theta = 30°, opposite side = 3, hypotenuse = 6: \sin 30° = \frac{3}{6} = 0.5

Notation

SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent

What This Formula Means

The three primary trigonometric ratios—sine, cosine, and tangent—defined as ratios of specific sides in a right triangle.

Imagine a ramp leaning against a wall. The steepness depends on the ratio of how high the wall is to how long the ramp is. Trigonometry gives names to these ratios: sine is how high compared to the ramp, cosine is how far along the ground compared to the ramp, and tangent is how high compared to how far along the ground. No matter how big or small the ramp, if the angle is the same, these ratios stay the same.

Formal View

For \triangle ABC with \angle C = \frac{\pi}{2}: \sin A = \frac{|BC|}{|AB|}, \cos A = \frac{|AC|}{|AB|}, \tan A = \frac{|BC|}{|AC|}; satisfies \sin^2 A + \cos^2 A = 1 and \tan A = \frac{\sin A}{\cos A}

Worked Examples

Example 1

easy
In a right triangle, the angle \theta = 30°, and the hypotenuse is 10. Find the lengths of the opposite and adjacent sides.

Solution

  1. 1
    Step 1: Recall the definitions: \sin\theta = \frac{\text{opp}}{\text{hyp}} and \cos\theta = \frac{\text{adj}}{\text{hyp}}.
  2. 2
    Step 2: Find the opposite side: \sin 30° = \frac{\text{opp}}{10}. Since \sin 30° = 0.5, we get \text{opp} = 10 \times 0.5 = 5.
  3. 3
    Step 3: Find the adjacent side: \cos 30° = \frac{\text{adj}}{10}. Since \cos 30° = \frac{\sqrt{3}}{2} \approx 0.866, we get \text{adj} = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66.

Answer

Opposite = 5, Adjacent = 5\sqrt{3}.
The sine ratio connects the opposite side to the hypotenuse, and the cosine ratio connects the adjacent side to the hypotenuse. For a 30° angle, these are well-known values: \sin 30° = \frac{1}{2} and \cos 30° = \frac{\sqrt{3}}{2}. Multiplying each by the hypotenuse length gives the side lengths.

Example 2

medium
A ladder 13 feet long leans against a wall. The base of the ladder is 5 feet from the wall. Find the angle the ladder makes with the ground (to the nearest degree).

Common Mistakes

  • Mixing up which side is opposite vs adjacent for the given angle
  • Applying trig ratios to non-right triangles without the law of sines/cosines
  • Forgetting that the hypotenuse is always the longest side, opposite the right angle

Why This Formula Matters

Used in navigation, engineering, physics, and any field that involves angles and distances—from measuring building heights to computing satellite orbits.

Frequently Asked Questions

What is the Right Triangle Trigonometry formula?

The three primary trigonometric ratios—sine, cosine, and tangent—defined as ratios of specific sides in a right triangle.

How do you use the Right Triangle Trigonometry formula?

Imagine a ramp leaning against a wall. The steepness depends on the ratio of how high the wall is to how long the ramp is. Trigonometry gives names to these ratios: sine is how high compared to the ramp, cosine is how far along the ground compared to the ramp, and tangent is how high compared to how far along the ground. No matter how big or small the ramp, if the angle is the same, these ratios stay the same.

What do the symbols mean in the Right Triangle Trigonometry formula?

SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent

Why is the Right Triangle Trigonometry formula important in Math?

Used in navigation, engineering, physics, and any field that involves angles and distances—from measuring building heights to computing satellite orbits.

What do students get wrong about Right Triangle Trigonometry?

Which side is 'opposite' and which is 'adjacent' depends on which angle you're looking at—they swap when you switch angles.

What should I learn before the Right Triangle Trigonometry formula?

Before studying the Right Triangle Trigonometry formula, you should understand: triangles, pythagorean theorem, ratios.

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