Educerie · SAT · Math
Geometry and Trigonometry · GT.3 Right triangles and trigonometry
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Use Pythagoras both ways | Find a side; test whether a triangle is right-angled |
| Choose sine, cosine or tangent | From which sides are given and which is wanted |
| Use the two special triangles | 45–45–90 and 30–60–90, both on the reference sheet |
| Use the complementary identity | sin(x°) = cos(90° − x°) |
| Convert degrees and radians | π radians = 180° |
1The idea in one paragraph
In a right triangle the ratio of any two sides is fixed by the angles, and sine, cosine and tangent are names for three of those ratios. Choosing between them is mechanical: look at which two sides the question involves relative to the angle you are working from. Nothing on this test requires the sine or cosine rule, and nothing goes beyond the unit circle.
SOH-CAH-TOA. sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Label the three sides relative to the angle before you choose.
2Pythagoras, and the triples worth recognising
a² + b² = c², with c the hypotenuse — always the side opposite the right angle, always the longest.
Recognising these saves time: 3-4-5, 5-12-13, 8-15-17, and every multiple (6-8-10, 9-12-15, 10-24-26). When a question gives two of a triple, the third is immediate.
Run it backwards to test a triangle: if a² + b² = c² the triangle is right-angled, and the test asks this as a yes/no.
3Choosing the ratio
Label relative to the angle you are using:
- Opposite — across from the angle
- Adjacent — next to it, and not the hypotenuse
- Hypotenuse — opposite the right angle
Then pick the ratio containing the side you know and the side you want. If you know the opposite and want the hypotenuse, that is sine. If you know both legs and want the angle, that is tangent, and you need its inverse.
4The special triangles
| Triangle | Sides in ratio |
|---|---|
| 45°–45°–90° | x : x : x√2 |
| 30°–60°–90° | x : x√3 : 2x — the shortest side faces 30° |
Both are printed on the reference sheet, and both appear on most forms. When a question's answer choices contain √2 or √3, the question is testing one of these and no calculator is needed.
5The complementary identity
sin(x°) = cos(90° − x°), and cos(x°) = sin(90° − x°).
It follows from the triangle itself: one angle's opposite side is the other angle's adjacent side. A question giving sin(x) = 0.6 and asking for cos(90° − x) is asking you to notice this and answer 0.6 without computing anything.
6Radians
π radians = 180°, so:
- Degrees to radians: × π/180
- Radians to degrees: × 180/π
The common values are worth knowing: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°. Arc length is rθ with θ in radians, which is why the radian form is used for circles.
Where points are lost
- Using the wrong side as the hypotenuse, which is always opposite the right angle.
- Choosing sine where the sides call for tangent.
- Forgetting the inverse function when the angle is the unknown.
- Reaching for a calculator on a special triangle that needs none.
- Missing the complementary identity and grinding through the triangle instead.
- Leaving the answer in degrees when radians were asked for.
- Assuming a triangle is right-angled because it looks it.
Work it right
- Find the right angle and label the hypotenuse.
- Label opposite and adjacent relative to the angle in play.
- Check for a Pythagorean triple or a special triangle first.
- Otherwise choose the ratio containing what you know and what you want.
- Use the inverse if the unknown is the angle, and check the units asked for.
Try it
Q1. A right triangle has legs of 9 and 12. What is the length of its hypotenuse? (Type your answer.)
Q2. In a right triangle, the angle θ has an opposite side of 7 and a hypotenuse of 25. What is cos θ?
A) 7/25 B) 24/25 C) 7/24 D) 25/24
Q3. If sin(x°) = 0.8, what is cos(90° − x°)?
A) 0.2 B) 0.6 C) 0.8 D) 1.25
Q4. In a 30°–60°–90° triangle, the side opposite the 30° angle is 5. What is the length of the hypotenuse? (Type your answer.)
Q5. What is 135° in radians?
A) 3π/4 B) 2π/3 C) 5π/6 D) 4π/3
In one breath
Label the hypotenuse first, then opposite and adjacent relative to the angle you are using, and pick the ratio containing what you know and what you want. Look for a Pythagorean triple or one of the two special triangles before reaching for the calculator — answer choices carrying √2 or √3 are telling you which one. And sin(x) = cos(90 − x) turns a calculation into a reading, because one angle's opposite side is the other's adjacent side.
Answers
Q1. 15. a 3-4-5 triangle, tripled
Q2. B — 24/25. find the third side first
A is sin θ; C is tan θ.
Q3. C — 0.8. the complementary identity cos(90° − x°) = sin(x°). B is the cosine of x itself for a 3-4-5 triangle, which is the distractor for a student who computes instead of noticing.
Q4. 10. in a 30–60–90 triangle the hypotenuse is twice the shortest side The side opposite 30° is the shortest, so the hypotenuse is 10. Answering 5√3 gives the side opposite 60°.
Q5. A — 3π/4. multiply by π/180
B is 120°, C is 150°, D is 240°.
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Geometry and Trigonometry, skill/knowledge testing point "Right triangles and trigonometry"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
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