A ratio describes an angle
Trigonometric ratios compare sides of a right triangle relative to one acute angle. Similar right triangles with that angle have the same ratios even when their sizes differ.
Learning goals
- Identify opposite, adjacent and hypotenuse relative to a chosen acute angle.
- Write sin, cos and tan as correct side ratios.
- Write cosec, sec and cot as reciprocal side ratios.
- Explain why ratios depend on angle rather than triangle size.
Concept lesson
Find the hypotenuse first
The hypotenuse is opposite the right angle and is unchanged when the reference acute angle changes. The other two sides are named opposite and adjacent relative to the selected angle.
Reciprocal means invert
The reciprocal of a/b is b/a. The reciprocal pairs are sin-cosec, cos-sec and tan-cot.
Side names change with the reference angle
Mark the reference angle. The side across from it is opposite; the non-hypotenuse side touching it is adjacent. Switching to the other acute angle swaps opposite and adjacent while the hypotenuse remains fixed.
Similarity makes the ratio invariant
Any two right triangles sharing an equal acute angle are similar by AA. Their corresponding sides have one scale factor, so each quotient of corresponding side types is unchanged. Therefore sin A, cos A and tan A depend on A, not triangle size.
Write primary ratios for either acute angle
For right triangle ABC right-angled at B, relative to A: sin A=BC/AC, cos A=AB/AC and tan A=BC/AB. Relative to C, opposite and adjacent swap: sin C=AB/AC, cos C=BC/AC and tan C=AB/BC.
Build the reciprocal ratios
Once sin A=opposite/hypotenuse, cosec A=hypotenuse/opposite. Similarly sec A=hypotenuse/adjacent and cot A=adjacent/opposite. Check each reciprocal product equals 1.
Switch the reference angle
For three rotated right triangles, label side roles for each acute angle and write sin and cos. Confirm sin of one acute angle equals cos of the other.
Keep hypotenuse out of tan
Correct proposed tan and cot ratios by first hiding the hypotenuse, then comparing only opposite and adjacent legs.
Reciprocal-pair check
Match six functions into reciprocal pairs, write both side ratios and verify each product is 1.
Choose a ratio for a ramp
Given a ramp angle and two named measurements, select sin, cos or tan by identifying which side types are known and required. Do not calculate beyond the supplied data.
Diagram-reading answer routine
Circle the reference angle, mark H/O/A, then write the requested ratio. This prevents label orientation from controlling the answer in one- and two-mark diagram questions.
Right-triangle ratios mastery
For two differently oriented right triangles, label side roles for both acute angles, write all six ratios for one angle, verify reciprocal pairs and explain why a scaled copy has equal ratios.