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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.
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Solving a Right Triangle
What is it?
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.
In simple words
- 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
Important terms
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).
What you should learn
- 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.
Examples
- a=15 sin 40
- b=15 cos 40
- Not 15 tan 40 for the hyp case
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).
Recommended for Class 10
Full school-level explanation
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.
Pick the ratio that uses what you have:
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.
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.
Remember
- opp = hyp sin
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).
Remember
- hyp = opp/sin
Examples
- c=50/sin 35.5
- c=sqrt(152+12.52)
- Check roughly 19.5
Remember
- sin opp/hyp
- Pythagoras on the right angle
- a = 15 sin 40
- name the angle
- opp = hyp sin
- hyp = opp/sin
- sin opp/hyp
- Pythagoras on the right angle
How the ideas connect
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
Deeper understanding
Go beyond memorising Solving a Right Triangle. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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 mistakes
Exam focus
Core Idea
ANGLE AND HYPOTENUSE
SIDE AND ANGLE, OR TWO SIDES
Worked Method
Exam focus
Quick revision
- sin opp/hyp
- Pythagoras on the right angle
- a = 15 sin 40
- name the angle
- SOLVING A RIGHT TRIANGLE: A right triangle has one 90 deg angle. The side opposite 90 deg is the hypotenuse. Given two sides,
- 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
Opposite and adjacent swap if you switch which acute angle you mean. Always name the angle first.
For long answers on Solving a Right Triangle: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- sin opp/hyp
- Pythagoras on the right angle
- a = 15 sin 40
- name the angle
Important formulas
- sin opp/hyp
- Pythagoras on the right angle
- a = 15 sin 40
Common exam traps
- Opposite and adjacent swap if you switch which acute angle you mean. Always name the angle first.
- Not 15 tan 40 for the hyp case
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Opposite and adjacent swap if you switch which acute angle you mean. Always name the angle first.
- Mistake
Not 15 tan 40 for the hyp case
- Mistake
Do not treat ANGLE AND HYPOTENUSE and SIDE AND ANGLE, OR TWO SIDES as the same step; each has a different purpose.
- Mistake
Opposite and adjacent swap if you switch which acute angle you mean. Always name the angle first.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
sin opp/hyp
Pythagoras on the right angle
a = 15 sin 40
name the angle
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
a = 15 sin 40
- 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
- Formula
- a = 15 sin 40
- Example
- a=15 sin 40
- Critical warning
- Opposite and adjacent swap if you switch which acute angle you mean. Always name the angle first.
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Exam Practice
Define
Q1. Define Solving a Right Triangle.
Explain
Q2. Explain: ANGLE AND HYPOTENUSE.
Example
Q3. Give one clear example of Solving a Right Triangle and say why it fits.
Mistake
Q4. State one common mistake students make about Solving a Right Triangle.
Exam note
Q5. Write a short exam note on Solving a Right Triangle: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.