Similarity & Trigonometry
GradesGeometry & FunctionsSimilarity & TrigonometryRight Triangle Trigonometry

Right Triangle Trigonometry

📐 Similarity & Trigonometry
📐

Right Triangle Trigonometry

Apply SOH-CAH-TOA to find sides and angles in right triangles.

In a right triangle, the three primary trigonometric ratios relate each acute angle to two of the three sides. These ratios — sine, cosine, and tangent — are constant for a given angle regardless of the triangle's size.

sin⁡θ=oppositehypotenusecos⁡θ=adjacenthypotenusetan⁡θ=oppositeadjacent\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}
📐

SOH-CAH-TOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

tan⁡θ=sin⁡θcos⁡θsin⁡2θ+cos⁡2θ=1\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \sin^2\theta + \cos^2\theta = 1
🏔️Finding a Missing Side
In right triangle ABC with ∠A = 35° and hypotenuse c = 20, find the side opposite ∠A. sin(35°) = opposite/20 → opposite = 20 · sin(35°) ≈ 20 · 0.574 ≈ 11.5.
sin⁡35∘=a20  ⟹  a=20sin⁡35∘≈11.5\sin 35^\circ = \frac{a}{20} \implies a = 20\sin 35^\circ \approx 11.5
💡

Remember This!

Use inverse trig functions to find angles: if sin θ = 0.6, then θ = sin⁻¹(0.6) ≈ 36.9°. Your calculator must be in degree mode for degree problems.

✏️ Try It!

In a right triangle, the opposite side is 7 and the adjacent side is 24. What is tan θ?

Practice with CalcVerse
🌊Trig Ratios (SOH‑CAH‑TOA)📐Special Right Triangles
Take Quiz 📝 — 25 Questions