Trig Ratios in a Right Triangle (quick List)

Trig Ratios in a Right Triangle (quick List)

What is it?

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.

In simple words

• 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

Important terms

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 = 35 then cos A = 45 in a 3-4-5 triangle (acute A). Then tan A = 34. sin²+cos² = 925+1625=1.

What you should learn

• 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.

Examples

• tan A = opp/adj
• sin A = opp/hyp
• cos A = adj/hyp

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.

Full school-level explanation

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, then reciprocal:

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.

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.

Remember

• opp/adj/hyp

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.

Remember

• 3-4-5 friend

Examples

• cos = 45
• tan=34
• Not sin=45 unless you swapped

Remember

• SOH-CAH-TOA
• sin²+cos²=1
• 35, 45, 34
• sin uses hyp
• opp/adj/hyp
• 3-4-5 friend
• SOH-CAH-TOA
• sin²+cos²=1

How the ideas connect

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

Deeper understanding

Go beyond memorising Trig Ratios in a Right Triangle (quick List). Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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

Common mistakes

Exam focus

Core Idea

PRIMARY

CHECKS

Worked Method

Exam focus

Quick revision

• SOH-CAH-TOA
• sin²+cos²=1
• 35, 45, 34
• sin uses hyp
• 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. I
• PRIMARY: Need hyp for sin and cos. Tan can skip the hyp. If you know two sides, you can find the third by Pyt

sin is never opposite/adjacent. That is tan. Mixing SOH and TOA is the usual ratio slip.

For long answers on Trig Ratios in a Right Triangle (quick List): define clearly → explain the main parts → give one example → state one exam caution.