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Right triangles and trigonometry: SAT Math notes

Find missing sides and angles in right triangles using the Pythagorean theorem, the special right triangles, and the sine, cosine and tangent ratios.

Study it with worked examples and 8 practice questionsPractise Right triangles and trigonometry questions

What this topic tests

Every question here is built on a right triangle, even when it's hidden inside a rectangle, an equilateral triangle or a bigger figure. Your first job is always the same: find the right angle, and with it the hypotenuse, the longest side, which sits across from the right angle.

With two sides known, the Pythagorean theorem gives you the third. With an angle and a side, or a ratio, sine, cosine and tangent connect them. Two special triangles, 45-45-90 and 30-60-90, have side patterns you can use with almost no calculation. The reference sheet gives the Pythagorean theorem and both special triangles, but not the trig ratios, so learn SOHCAHTOA.

Harder questions often test one neat fact: the two acute angles of a right triangle add to 90°, so the sine of one equals the cosine of the other. Similar triangles also come up a lot, because similar triangles have the same angles and so the same trig ratios.

Key ideas

The 4 kinds of Right triangles and trigonometry question

Sorted from every Right triangles and trigonometry question in the site's bank: each kind keeps coming up, and has a method that works every time.

1. The Pythagorean theorem

How to spot it. You know two sides of a right triangle and need the third. Watch for a rectangle with a diagonal: the diagonal splits it into two right triangles.

How to answer it:

  1. Find the right angle. The side across from it is the hypotenuse, c.
  2. Write a2+b2=c2 with the numbers in the right places.
  3. If the missing side is a leg, subtract: b2=c2-a2.
  4. Square any roots carefully: (5√(17))2=25×17.
  5. Take the square root at the end, and check that the hypotenuse is still the longest side.

The trap. Adding the squares when the missing side is a leg. If the longest side is given, subtract instead. Another slip is squaring only part of a term like 3√(17): square the 3 and the root.

2. Sine, cosine and tangent (SOHCAHTOA)

How to spot it. The question mentions sin, cos or tan. It asks for a ratio from the side lengths, or gives a ratio and asks for a side. Sometimes the triangle is only described in words, or it's similar to another triangle.

How to answer it:

  1. Find the right angle and the hypotenuse. Then, from the angle in the question, label the other two sides: opposite (across from the angle) and adjacent (touching the angle, but not the hypotenuse).
  2. Pick the ratio you need with SOHCAHTOA.
  3. If a ratio is given, like cos A=3/5, treat it as sides 3k and 5k. Find the third side with the Pythagorean theorem, then use a side you know to find k.
  4. If the triangle is similar to another one, switch to the matching angle. Its ratios are the same.
  5. Write the ratio or the length, and simplify the fraction.

The trap. Labelling the sides from the wrong angle. Opposite and adjacent depend on which angle you stand at: the side opposite A is adjacent to the other acute angle. Name the angle first, then label the sides.

3. Complementary angles: sine of one, cosine of the other

How to spot it. The question says the sine of one angle equals the cosine of another, compares the two acute angles of a right triangle, or gives the tangent of one acute angle and asks about the other.

How to answer it:

  1. Remember why it works: in a right triangle, the side opposite one acute angle is adjacent to the other. So sin A=cos B whenever A+B=90°.
  2. If you're told sin A=cos B for acute angles, write A+B=90 (or (π)/2 in radians) and solve.
  3. For tangent, flip the fraction: the two acute angles have reciprocal tangents.
  4. If the angles are given as expressions like x+52, add them, set the total equal to 90, and solve for x.

The trap. Setting the angles equal. sin A=cos B doesn't mean A=B; it means they add to 90°. For tangent, the other angle's tangent is the reciprocal, not the same value and not the negative.

4. Special right triangles

How to spot it. You see angles of 45°, 30° or 60°, an isosceles right triangle, an equilateral triangle, or answer choices with √(2) or √(3) in them.

How to answer it:

  1. Decide which triangle it is. Two equal legs or a 45° angle means 45-45-90. A 30° or 60° angle, or half of an equilateral triangle, means 30-60-90.
  2. Write the side pattern from the reference sheet with one unknown: s, s, s√(2) or x, x√(3), 2x.
  3. Match what you're given to the pattern. It might be one side, or a perimeter or height written in terms of the unknown.
  4. Solve for the unknown, then find the side the question asks for.
  5. For an equilateral triangle, the height cuts it into two 30-60-90 triangles, so the height is (√(3))/2 times the side.

The trap. Mixing up which side goes with which angle in a 30-60-90 triangle. The shortest side x is opposite 30°, x√(3) is opposite 60°, and 2x, the hypotenuse, is opposite the right angle.

Practise Right triangles and trigonometry

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