O Level MathematicsC6.2 Right-angled trigonometry (SOH CAH TOA).

🔺 Mastering SOH CAH TOA: Crack Right-Angle Trig in Minutes!

Edudent Academy
17 Jan 26

Right-angled trigonometry is a **must-know** tool for solving angles and side lengths in right-angled triangles. From navigation questions to exam diagrams, understanding SOH CAH TOA can earn you easy marks and build confidence for harder topics like bearings and vectors.

The Core Idea: SOH CAH TOA

Remember the three famous ratios: **Sine = Opposite / Hypotenuse**, **Cosine = Adjacent / Hypotenuse**, and **Tangent = Opposite / Adjacent**. In symbols: sin⁡θ=OH, cos⁡θ=AH, tan⁡θ=OA\displaystyle \sin\theta = \frac{\text{O}}{\text{H}}, \ \cos\theta = \frac{\text{A}}{\text{H}}, \ \tan\theta = \frac{\text{O}}{\text{A}}. Identify the reference angle θ\theta, label the sides correctly, choose the right ratio, and rearrange.

  • Key point 1: The **hypotenuse** is always the longest side opposite the right angle.
  • Key point 2: Use the **inverse functions** sin⁡−1\sin^{-1}, cos⁡−1\cos^{-1}, or tan⁡−1\tan^{-1} when you need to find an angle.

Worked Example: Ramp Inclination

Problem: A wheelchair ramp rises 0.8 m over a horizontal distance of 4 m. Find (a) the length of the ramp, (b) the angle of elevation of the ramp.

  • Step 1:
    Length (hypotenuse)=0.82+42=0.64+16=16.64≈4.08 m\text{Length (hypotenuse)} = \sqrt{0.8^2 + 4^2} = \sqrt{0.64 + 16} = \sqrt{16.64} \approx 4.08\,\text{m}
  • Step 2:
    tan⁡θ=0.84  ⇒  θ=tan⁡−1 ⁣(0.84)≈11.3∘\tan\theta = \frac{0.8}{4} \;\Rightarrow\; \theta = \tan^{-1}\!\left(\frac{0.8}{4}\right) \approx 11.3^{\circ}

With systematic practice and careful side-labeling, **SOH CAH TOA becomes second nature**. Tackle a mix of textbook and past-paper questions to solidify your skills—your exam marks will thank you!