Right Triangle Trigonometry Formula

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

The Formula

sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\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 θ=30°\theta = 30°, opposite side =3= 3, hypotenuse =6= 6: sin⁡30°=36=0.5\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 △ABC\triangle ABC with ∠C=π2\angle C = \frac{\pi}{2}: sin⁡A=∣BC∣∣AB∣\sin A = \frac{|BC|}{|AB|}, cos⁡A=∣AC∣∣AB∣\cos A = \frac{|AC|}{|AB|}, tan⁡A=∣BC∣∣AC∣\tan A = \frac{|BC|}{|AC|}; satisfies sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1 and tan⁡A=sin⁡Acos⁡A\tan A = \frac{\sin A}{\cos A}

Worked Examples

Example 1

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

Answer

Opposite =5= 5, Adjacent =53= 5\sqrt{3}.

First step

1
Step 1: Recall the definitions: sin⁡θ=opphyp\sin\theta = \frac{\text{opp}}{\text{hyp}} and cos⁡θ=adjhyp\cos\theta = \frac{\text{adj}}{\text{hyp}}.

Full solution

  1. 2
    Step 2: Find the opposite side: sin⁡30°=opp10\sin 30° = \frac{\text{opp}}{10}. Since sin⁡30°=0.5\sin 30° = 0.5, we get opp=10×0.5=5\text{opp} = 10 \times 0.5 = 5.
  2. 3
    Step 3: Find the adjacent side: cos⁡30°=adj10\cos 30° = \frac{\text{adj}}{10}. Since cos⁡30°=32≈0.866\cos 30° = \frac{\sqrt{3}}{2} \approx 0.866, we get adj=10×32=53≈8.66\text{adj} = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66.
The sine ratio connects the opposite side to the hypotenuse, and the cosine ratio connects the adjacent side to the hypotenuse. For a 30°30° angle, these are well-known values: sin⁡30°=12\sin 30° = \frac{1}{2} and cos⁡30°=32\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).

Example 3

medium
A 24-foot ladder leans against a wall at an angle of 75°75° to the ground. How high up the wall does it reach (to the nearest tenth of a foot)?

Common Mistakes

  • Mislabeling opposite and adjacent — fix them relative to the chosen angle, not to the page.
  • Using sine when the two known sides are the legs — that pairing (opposite over adjacent) is tangent.
  • Forgetting the calculator's angle mode — set degrees for degree problems before evaluating sine, cosine, or tangent.

Why This Formula Matters

It is how angles enter measurement: heights of buildings, ramp steepness, and navigation all come from one angle and one side. Knowing which sides each ratio uses — relative to the chosen angle — is what separates trig from the Pythagorean theorem, which never uses an angle. Recognizing it by "Is there a right angle and an acute angle linking a pair of sides I need to relate?" — rather than by familiar numbers — is what lets a student tell it apart from pythagorean theorem and special right triangles and inverse trig functions in a mixed problem set.

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?

It is how angles enter measurement: heights of buildings, ramp steepness, and navigation all come from one angle and one side. Knowing which sides each ratio uses — relative to the chosen angle — is what separates trig from the Pythagorean theorem, which never uses an angle. Recognizing it by "Is there a right angle and an acute angle linking a pair of sides I need to relate?" — rather than by familiar numbers — is what lets a student tell it apart from pythagorean theorem and special right triangles and inverse trig functions in a mixed problem set.

What do students get wrong about Right Triangle Trigonometry?

The procedure for right triangle trigonometry is the easy part; the trap is mislabeling opposite and adjacent. Asking "Is there a right angle and an acute angle linking a pair of sides I need to relate?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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