MathematicsChapter 8: Application of Trigonometry
Trig Ratios in A Right Triangle (quick List)
sin A = opp/hyp, cos A = adj/hyp, tan A = opp/adj.
Learning objectives
- Explain trig ratios in a right triangle (quick list) using the correct definitions, conditions, and notation.
- Apply SOH-CAH-TOA, sin^2+cos^2=1 correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
sin A = opp/hyp, cos A = adj/hyp, tan A = opp/adj. Reciprocals: cosec=1/sin, sec=1/cos, cot=1/tan. Identity tan A = sin A / cos A. Pythagoras: sin^2 A + cos^2 A = 1. These identities check a ratio pair.
Mathematical form
hyp = 5
sin = 35
cos = 45
tan = 34
Where / symbols
- TRIG RATIOS IN A RIGHT TRIANGLE (QUICK LIST)
- sin A = opp/hyp, cos A = adj/hyp, tan A = opp/adj. Reciprocals: cosec=1/sin, sec=1/cos, cot=1/tan. Identity tan A = sin A / cos A. Pythagoras: sin^2 A + cos^2 A = 1. These identities check a ratio pair.
- PRIMARY
- Need hyp for sin and cos. Tan can skip the hyp. If you know two sides, you can find the third by Pythagoras, then any ratio.
- CHECKS
- If sin A = 3/5 then cos A = 4/5 in a 3-4-5 triangle (acute A). Then tan A = 3/4. sin^2+cos^2 = 9/25+16/25=1.
Relationship
- sin2+cos2=1
Overview
Primary, then reciprocal:
Primary
Need hyp for sin and cos. Tan can skip the hyp. If you know two sides, you can find the third by Pythagoras, then any ratio.
Examples tan A = opp/adj · sin A = opp/hyp · cos A = adj/hyp
Checks
If sin A = 35 then cos A = 45 in a 3-4-5 triangle (acute A). Then tan A = 34. sin2+cos2 = 925+1625=1.
Examples cos = 4/5 · tan=3/4 · Not sin=4/5 unless you swapped
WHAT — TRIG RATIOS IN A RIGHT TRIANGLE (QUICK LIST)
sin A = opp/hyp, cos A = adj/hyp, tan A = opp/adj. Reciprocals: cosec=1/sin, sec=1/cos, cot=1/tan. Identity tan A = sin A / cos A. Pythagoras: sin^2 A + cos^2 A = 1. These identities check a ratio pair. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Trig Ratios In A Right Triangle (Quick List) matters because trigonometry links angles and lengths. It lets us solve triangles and practical height, distance, bearing, and area problems without measuring every side directly.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through PRIMARY, then CHECKS. Worked: right triangle, opp = 3, adj = 4. Find sin, cos, tan of that angle. hyp^2 = 3^2 + 4^2 = 9 + 16 = 25 hyp = 5 sin = 35 cos = 45 tan = 34 Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- right triangle, opp = 3, adj = 4. Find sin, cos, tan of that angle.
- hyp2 = 32 + 42 = 9 + 16 = 25
- hyp = 5
- sin = 35
- cos = 45
- tan = 34
In simple words
sin is never opposite/adjacent. That is tan. Mixing SOH and TOA is the usual ratio slip.
Key points
- SOH-CAH-TOA
- sin2+cos2=1
- opp/adj/hyp
- 3-4-5 friend
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Trig Ratios in A Right Triangle (quick List)
Sticky Notes · Quick Revision
Core Idea
- sin A = opp/hyp, cos A = adj/hyp, tan A = opp/adj. Reciprocals: cosec=1/sin, sec=1/cos, cot=1/tan. Identity tan A = sin A / cos A. Pythagoras: sin^2 A + cos^2 A = 1. These identities check a ratio pair.
PRIMARY
- Need hyp for sin and cos. Tan can skip the hyp. If you know two sides, you can find the third by Pythagoras, then any ratio.
CHECKS
- If sin A = 35 then cos A = 45 in a 3-4-5 triangle (acute A). Then tan A = 34. sin2+cos2 = 925+1625=1.
Worked Method
- Worked: right triangle, opp = 3, adj = 4. Find sin, cos, tan of that angle.
- hyp2 = 32 + 42 = 9 + 16 = 25
- hyp = 5
- sin = 35
- cos = 45
- tan = 34
Common Misconception
- sin is never opposite/adjacent. That is tan. Mixing SOH and TOA is the usual ratio slip.
FBISE Exam Check
- sin is never opposite/adjacent. That is tan. Mixing SOH and TOA is the usual ratio slip.
60-second revision
- sin A = opp/hyp, cos A = adj/hyp, tan A = opp/adj.
- PRIMARY: Need hyp for sin and cos.
- CHECKS: If sin A = 35 then cos A = 45 in a 3-4-5 triangle (acute A).
- Worked-method reminder: tan = 34
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