One ratio can determine all six
For an acute angle, one ratio fixes two relative side lengths. Pythagoras supplies the third side, after which all six ratios follow from the same H/O/A labels.
Learning goals
- Construct a valid right triangle from one given trigonometric ratio.
- Use Pythagoras theorem to obtain the missing side.
- Determine all other ratios accurately.
- Use reciprocal and quotient relations efficiently.
Concept lesson
Translate functions into side pairs
sin=O/H, cos=A/H, tan=O/A; cosec=H/O, sec=H/A, cot=A/O. Translate the given function before assigning any numbers.
Find a missing leg, not a hypotenuse
If H is known, a missing leg satisfies leg=sqrt(H^2-other leg^2). If both legs are known, H=sqrt(O^2+A^2).
Construct a ratio triangle
For sin A=3/5, assign O=3k and H=5k; for tan A=8/15, assign O=8k and A=15k. A common positive scale k does not affect any ratio and may be set to 1 for calculation.
Two solution routes
Use a triangle route when a missing side or several ratios are required. Use reciprocal and quotient relations for a direct conversion such as cosec=1/sin, sec=1/cos, cot=1/tan, tan=sin/cos and cot=cos/sin.
Recover the missing side
If sin A=3/5, take O=3 and H=5. Then A=sqrt(5^2-3^2)=4. The result is valid because the hypotenuse is the longest side.
Generate all six ratios
From O=8, A=15 and H=17, write sin=8/17, cos=15/17, tan=8/15, cosec=17/8, sec=17/15 and cot=15/8. Verify the three reciprocal products equal 1.
Can this fraction be the stated ratio?
Check proposed side assignments. Reject any sin or cos construction with a leg longer than its hypotenuse, and reject assignments inconsistent with the function's side pair.
Direct conversion without rebuilding
Convert sin to cosec, cos to sec and cot to tan by reciprocation; convert a supplied sin and cos pair into tan by division. State which relation was used.
Freeze H/O/A before filling the table
Label H/O/A once, calculate the missing side, complete a six-ratio table, and check reciprocal and quotient consistency.
Determine a slope angle ratio
A support has known rise and sloping length. Translate these to O and H, find the horizontal run with Pythagoras, then report all ratios for the base angle.
Show the recover-and-convert chain
For two marks, show ratio-to-side assignment, Pythagorean recovery and the requested final ratio. For a one-step reciprocal question, name and apply the reciprocal relation directly.
Other-ratios mastery
From one supplied ratio, build a valid right triangle, recover the missing side, find all other ratios and solve two direct reciprocal or quotient conversions with consistency checks.