Finding Missing Angles

Sometimes you know two sides but not the angle. You first make a decimal ratio (for example $0.5$), then ask: “Which angle has sine equal to $0.5$?”

That “undo” step is called the inverse.

Important — sin⁻¹ is NOT $\frac{1}{\sin }$. On a calculator the inverse buttons are usually marked sin⁻¹, cos⁻¹, tan⁻¹ (or arcsin, arccos, arctan). Press that button after you have the decimal ratio.

• 1Sketch and label O, A, H for the angle you want (call it θ).
• 2Make a ratio from the two known sides (a single decimal or exact fraction).
• 3Match the ratio to sin, cos, or tan (SOH CAH TOA).
• 4Apply the inverse on a calculator (degree mode), or recognise a value from the exact-values table.

Quick matching guide:

• Know Opposite and Hypotenuse → $\sin (\theta )=\frac{O}{H}$ → $\theta ={\sin }^{-1}\left(\frac{O}{H}\right)$
• Know Adjacent and Hypotenuse → $\cos (\theta )=\frac{A}{H}$ → $\theta ={\cos }^{-1}\left(\frac{A}{H}\right)$
• Know Opposite and Adjacent → $\tan (\theta )=\frac{O}{A}$ → $\theta ={\tan }^{-1}\left(\frac{O}{A}\right)$

If your ratio matches a table entry, you can read the angle without a calculator:

• Ratio $\frac{1}{2}$ as a sine → $30$° (because $\sin (30^{\circ })=\frac{1}{2}$)
• Ratio $\frac{1}{2}$ as a cosine → $60$° (because $\cos (60^{\circ })=\frac{1}{2}$)
• Ratio $1$ as a tangent → $45$° (because $\tan (45^{\circ })=1$)

If the decimal is not in the table (for example $0.75$), use ${\tan }^{-1}$ (or sin⁻¹ / cos⁻¹) on the calculator.