Right triangle trigonometry
Trigonometry on this test stays inside right triangles. There are three ratios, one identity worth knowing, and a handful of triangles that appear so often they are worth recognising on sight. The rest is labelling the triangle correctly before writing anything.
Label against the angle, not the picture
Opposite and adjacent are decided by which acute angle the question names, never by where the sides happen to sit on the page. The hypotenuse is always opposite the right angle. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. Students who lose marks here almost always used the other acute angle's opposite side.
Find the third side first
Most ratio questions give you two sides and ask for a ratio involving the third. Use the Pythagorean theorem before touching sine or cosine. Certain triples appear constantly and are worth memorising, because recognising one saves the arithmetic and gives you exact numbers.
- 3, 4, 5 and its multiples, such as 6, 8, 10
- 5, 12, 13 and its multiples, such as 10, 24, 26
- 8, 15, 17
- 7, 24, 25
The two special triangles
A 45, 45, 90 triangle has sides in the ratio 1 to 1 to root 2, so the hypotenuse is the leg times root 2. A 30, 60, 90 triangle has sides in the ratio 1 to root 3 to 2, with the short side opposite the 30 degree angle and the hypotenuse twice it. These ratios produce the exact values the test expects, which is why questions ask for them without offering a calculator answer.
The complementary angle identity
In a right triangle the two acute angles add to 90 degrees, so the side opposite one is the side adjacent to the other. That makes sine of A equal cosine of 90 minus A. Questions written as sine of 37 equals cosine of x are testing exactly this, and the answer is 53 without any trigonometry at all.
Exact answers and decimal answers
If the angle is one of the special ones, the answer is exact, and an option like root 3 over 2 is the one to pick. If the angle is not special and the question says round to the nearest tenth, the calculator is expected and a decimal is correct. Reading which of the two the question wants is part of the work, because the wrong form is usually sitting among the options.
Building a triangle from a ratio
When a question gives tangent and asks for sine, do not reach for an identity. Draw a right triangle with legs in the ratio you were given, find the hypotenuse, and read the sine off the picture. Tangent of 3/4 means legs 3 and 4, a hypotenuse of 5, and a sine of 3/5. This move also handles the reverse direction, from sine back to tangent.
How do I know which side is opposite?
The side that does not touch the named angle, other than the hypotenuse, is opposite. The side that touches it is adjacent. The hypotenuse is always across from the right angle.
What are the exact values for 30, 45 and 60 degrees?
Sine of 30 is 1/2, sine of 45 is root 2 over 2, sine of 60 is root 3 over 2. Cosine runs the same list backwards, and tangent of 45 is 1.
In a right triangle, if sin A = 0.6, what is cos A?
0.8. A sine of 3/5 means a 3, 4, 5 triangle, so the adjacent side is 4 and the cosine is 4/5.
Why does sin A equal cos of 90 minus A?
Because the two acute angles in a right triangle are complementary, so the side opposite one angle is the side adjacent to the other. The ratio swaps roles, which is exactly what the identity says.
Related guides
Updated 2026-10-25. Free to use, no account needed. SAT is a trademark of College Board. Not affiliated with or endorsed by College Board.