MathematicsChapter 8: Application of Trigonometry
Solving A Right Triangle
A right triangle has one 90 deg angle.
Learning objectives
- Explain solving a right triangle using the correct definitions, conditions, and notation.
- Apply sin opp/hyp, Pythagoras on the right angle correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
A right triangle has one 90 deg angle. The side opposite 90 deg is the hypotenuse. Given two sides, or one side and one acute angle, you can solve the rest with sin=opp/hyp, cos=adj/hyp, tan=opp/adj, and Pythagoras. Label opposite and adjacent relative to the acute angle you are using.
Mathematical form
a = 15 sin 40 deg
sin=opp/hyp
cos=adj/hyp
tan=opp/adj
Where / symbols
- SOLVING A RIGHT TRIANGLE
- A right triangle has one 90 deg angle. The side opposite 90 deg is the hypotenuse. Given two sides, or one side and one acute angle, you can solve the rest with sin=opp/hyp, cos=adj/hyp, tan=opp/adj, and Pythagoras. Label opposite and adjacent relative to the acute angle you are using.
- ANGLE AND HYPOTENUSE
- Angle A=40 deg, hyp c=15 m: side a opposite A is 15 sin 40. Adjacent b is 15 cos 40. Do not swap sin and cos.
- SIDE AND ANGLE, OR TWO SIDES
- If opposite b=50 and B=35.5 deg, hyp = 50 / sin 35.5. Two legs 15 and 12.5: hyp = sqrt(15^2+12.5^2).
Overview
Pick the ratio that uses what you have:
Angle and Hypotenuse
Angle A=40 deg, hyp c=15 m: side a opposite A is 15 sin 40. Adjacent b is 15 cos 40. Do not swap sin and cos.
Examples a=15 sin 40 · b=15 cos 40 · Not 15 tan 40 for the hyp case
Side and Angle, Or Two Sides
If opposite b=50 and B=35.5 deg, hyp = 50 / sin 35.5. Two legs 15 and 12.5: hyp = sqrt(15^2+12.5^2).
Examples c=50/sin 35.5 · c=sqrt(15^2+12.5^2) · Check roughly 19.5
WHAT — SOLVING A RIGHT TRIANGLE
A right triangle has one 90 deg angle. The side opposite 90 deg is the hypotenuse. Given two sides, or one side and one acute angle, you can solve the rest with sin=opp/hyp, cos=adj/hyp, tan=opp/adj, and Pythagoras. Label opposite and adjacent relative to the acute angle you are using. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Solving A Right Triangle 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 ANGLE AND HYPOTENUSE, then SIDE AND ANGLE, OR TWO SIDES. Worked: right-angled at C, A = 40 deg, hypotenuse c = 15 m. Find a (opposite A). sin A = opposite / hypotenuse sin 40 = a / 15 a = 15 sin 40 deg Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- right-angled at C, A = 40 deg, hypotenuse c = 15 m. Find a (opposite A).
- sin A = opposite / hypotenuse
- sin 40 = a / 15
- a = 15 sin 40 deg
In simple words
Opposite and adjacent swap if you switch which acute angle you mean. Always name the angle first.
Key points
- sin opp/hyp
- Pythagoras on the right angle
- opp = hyp sin
- hyp = opp/sin
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Solving A Right Triangle
Sticky Notes · Quick Revision
Core Idea
- A right triangle has one 90 deg angle. The side opposite 90 deg is the hypotenuse. Given two sides, or one side and one acute angle, you can solve the rest with sin=opp/hyp, cos=adj/hyp, tan=opp/adj, and Pythagoras. Label opposite and adjacent relative to the acute angle you are using.
ANGLE AND HYPOTENUSE
- Angle A=40 deg, hyp c=15 m: side a opposite A is 15 sin 40. Adjacent b is 15 cos 40. Do not swap sin and cos.
SIDE AND ANGLE, OR TWO SIDES
- If opposite b=50 and B=35.5 deg, hyp = 50 / sin 35.5. Two legs 15 and 12.5: hyp = sqrt(15^2+12.5^2).
Worked Method
- Worked: right-angled at C, A = 40 deg, hypotenuse c = 15 m. Find a (opposite A).
- sin A = opposite / hypotenuse
- sin 40 = a / 15
- a = 15 sin 40 deg
Common Misconception
- Opposite and adjacent swap if you switch which acute angle you mean. Always name the angle first.
FBISE Exam Check
- Opposite and adjacent swap if you switch which acute angle you mean. Always name the angle first.
60-second revision
- A right triangle has one 90 deg angle.
- ANGLE AND HYPOTENUSE: Angle A=40 deg, hyp c=15 m: side a opposite A is 15 sin 40.
- SIDE AND ANGLE, OR TWO SIDES: If opposite b=50 and B=35.5 deg, hyp = 50 / sin 35.5.
- Worked-method reminder: a = 15 sin 40 deg
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