SAT Math · Geometry and Trigonometry · Skill 3 of 4

Right triangles and trigonometry

Use the Pythagorean theorem and the two special right triangles, then sine, cosine and tangent to find any missing side or angle, by hand or in Desmos.

9 chapters plus a challengeAbout 40 minutesFree, no sign-up
Start with the ratio lab →Jump to the challenge
Hi, I’m Io. Trigonometry sounds scary, but it is just three fractions. Let me show you why they never change for the same angle.
Tap Io

Same angle, same ratios

Explore · 5 min

Here is a right triangle with angle A at the bottom left. The sides are named from A’s point of view: opposite (across from A), adjacent (next to A) and the hypotenuse (across from the right angle, always the longest). Resize the triangle, then change the angle.

The Pythagorean theorem

Practise · 5 min

In any right triangle, the two shorter sides (the legs) and the hypotenuse c are linked:

a² + b² = c²
  • Finding the hypotenuse: add the squares, then take the square root
  • Finding a leg: subtract, c² − a², then take the square root
  • Pythagorean triples save time: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and any multiple such as 6-8-10

The two special right triangles

Learn · 5 min

45°-45°-90°Legs x and x, hypotenuse x√2. Half a square, cut along the diagonal
30°-60°-90°Short leg x, long leg x√3, hypotenuse 2x. Half an equilateral triangle

Answers with √When a box is followed by √2 or √3, type only the number in front of it.

Sine, cosine and tangent

Practise · 6 min

For an acute angle in a right triangle, the three ratios are:

SOHsin A = opposite ÷ hypotenuse
CAHcos A = adjacent ÷ hypotenuse
TOAtan A = opposite ÷ adjacent
Worked example: in right triangle ABC, angle C is the right angle, BC = 8, AC = 15 and AB = 17. Find sin A, cos A and tan A.
  1. From A: the opposite side is BC = 8, the adjacent side is AC = 15, and the hypotenuse is AB = 17
  2. sin A = 8/17, cos A = 15/17, tan A = 8/15
  3. From angle B the roles swap: sin B = 15/17 = cos A

The sine and cosine link

Learn · 4 min

In a right triangle, the two acute angles add to 90°. The side opposite one of them is adjacent to the other, so:

sin x° = cos(90 − x)°

The SAT tests this directly: if sin A = cos B in a right triangle, then A + B = 90.

Which ratio should I use?

Sort · 4 min

Name the side you know and the side you want, both from the angle you are given. The pair decides the ratio: opposite and hypotenuse means sine, adjacent and hypotenuse means cosine, opposite and adjacent means tangent.

Trigonometry in Desmos

Tools · 4 min

When the angle is not special, use the built-in Desmos calculator. Press play on each card to see what to type.

Right triangles in real life

Scenarios · 6 min

Each tab is a situation you could meet on test day. Answer every question to finish the chapter.

Spot the trap

Detective · 3 min

Each solution has one faulty step. All three slips appear as wrong answer choices on the SAT.

Final challenge

8 questions · 10 min

Key ideas to carry in
Pythagorasa² + b² = c², where c is the hypotenuse.
45-45-90x, x, x√2.
30-60-90x, x√3, 2x.
SOHCAHTOAsin = O/H, cos = A/H, tan = O/A.
Complementssin x° = cos(90 − x)°.
DesmosDegree mode. arcsin, arccos, arctan give angles.
Challenge