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Right Triangles and Trigonometry
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Spot the triple
- Start:
- Look at the two given sides.
- Stop:
- Stop as soon as they match 3-4-5, 5-12-13, 8-15-17 or 7-24-25 or a multiple.
- Skip:
- Skip squaring and square roots.
- Why:
- A triple gives the answer in seconds and avoids arithmetic errors.
Used in questions 1, 3, 7
2. Hypotenuse is longest
- Start:
- Find which side is across from the right angle.
- Stop:
- Stop when that side is labelled c.
- Skip:
- Skip the other labels until c is fixed.
- Why:
- Putting a leg in the c place gives a wrong answer that is often one of the options.
Used in questions 1, 3, 7
3. SOH-CAH-TOA from the angle
- Start:
- Point at the angle in the question and label opposite, adjacent, hypotenuse.
- Stop:
- Stop when you write the fraction.
- Skip:
- Skip other angles.
- Why:
- Labels change with the angle, so labelling first prevents swapping sides.
Used in questions 2, 4, 8
4. Ratio to triangle
- Start:
- Turn a trig ratio into side numbers like 3, 4, 5.
- Stop:
- Stop when you know the scale factor from the side you are given.
- Skip:
- Skip decimals and angle sizes.
- Why:
- The ratio fixes the shape, and only the size changes by one scale number.
Used in questions 4, 5, 8
5. Sin equals cos of the other angle
- Start:
- Check if the two angles in the question are the acute angles of one right triangle.
- Stop:
- Stop when you write sin A = cos B.
- Skip:
- Do not calculate any side.
- Why:
- Complementary angles swap opposite and adjacent, so sin and cos swap too.
Used in question 5
6. Square diagonal, quick area
- Start:
- Divide the diagonal by √2 to get the side.
- Stop:
- Stop when you have the side, then square it.
- Skip:
- Do not square the diagonal and forget to halve.
- Why:
- The area is half of the diagonal squared, which is a quick check.
Used in question 6
7. Hunt for the hidden right angle
- Start:
- Check whether the three given sides form a triple (3-4-5, 5-12-13, 7-24-25, or a multiple) even if no right angle is drawn.
- Stop:
- Stop as soon as the squares add up (a² + b² = c²).
- Skip:
- Skip hard algebra; area and ratios are easy once you know the angle is 90°.
- Why:
- In question 16 the right angle is not marked, but 15-20-25 is a 3-4-5 triple, and then the area gives the altitude 12 quickly.
Used in questions 16, 11, 12
8. Sin and cos swap for the other angle
- Start:
- Look for sin or cos of two angles in one equation.
- Stop:
- Stop when you write that the two angles add to 90°.
- Skip:
- Do not try to find the angle by guess or calculator.
- Why:
- sin of an angle equals cos of its complement. So (3x + 5) + (2x + 10) = 90 solves question 14 in one line.
Used in questions 14, 10
9. Add three squares in a box
- Start:
- Write the three edge lengths of the box.
- Stop:
- Stop after √(a² + b² + c²).
- Skip:
- Skip drawing the base diagonal as a separate step.
- Why:
- The space diagonal is two Pythagoras steps in one, and 12² + 4² + 3² = 169 gives 13 right away.
Used in questions 15, 9