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Trigonometry has a reputation, and the SAT does not deserve credit for it. SAT trig stays inside right triangles: one theorem, three ratios, two special triangles, one heavily tested identity. None of it is magic. Sine, cosine, and tangent are names for side ratios, and once you label the sides relative to the angle, every question becomes a fraction you can write down.
SAT trigonometry is right-triangle trig only: the Pythagorean theorem, the sine, cosine, and tangent ratios (SOH-CAH-TOA), the two special triangles from the reference sheet, and the identity that sine of an angle equals cosine of its complement. The unit circle and trig graphs are not tested, so the whole topic fits on one page.
Key takeaways
- Pythagorean theorem: a^2 + b^2 = c^2, where c is the hypotenuse, always the side opposite the right angle.
- Recognize the families: 3-4-5 and 5-12-13, plus their multiples like 6-8-10 and 10-24-26.
- SOH-CAH-TOA: \sin = \tfrac{\text{opp}}{\text{hyp}}, \cos = \tfrac{\text{adj}}{\text{hyp}}, \tan = \tfrac{\text{opp}}{\text{adj}}. Ratios, fixed by the angle.
- The identity: \sin(x^\circ) = \cos(90^\circ - x^\circ). When \sin of one angle equals \cos of another, the angles sum to 90.
The Pythagorean theorem and the triangles worth recognizing
In any right triangle, the legs and hypotenuse obey a^2 + b^2 = c^2. The SAT reuses a few integer triangles constantly: 3-4-5 and 5-12-13, their scaled copies (6-8-10, 10-24-26), and the occasional 8-15-17. Recognizing a family turns a square-root computation into a lookup. See legs of 5 and 12, write 13, move on. For the angle relationships that sit beside these side rules, see angles and triangles.
The hypotenuse is the side opposite the right angle, full stop (also always the longest side). Given sides 6 and 10 where 10 is the hypotenuse, the missing leg is \sqrt{100 - 36} = 8, not \sqrt{136}. Plugging the hypotenuse in as a leg is the most common Pythagorean error.
SOH-CAH-TOA: three fractions with names
For an acute angle in a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. The deep fact is that these ratios depend only on the angle: every right triangle with that angle is a scaled copy of the same shape, so the fractions never change. The practical move is to label first. Mark the angle you care about, then write opp, adj, and hyp on the sides before touching the ratio.
Try one
In right triangle ABC, the right angle is at C, AC = 15, BC = 8, and AB = 17. What is the value of \tan A?
The special triangles (they are on the reference sheet)
Two triangles come with fixed side ratios, and both are drawn on the digital SAT’s reference sheet, so this is recognition, not recall. A 45-45-90 triangle has legs x and hypotenuse x\sqrt{2}. A 30-60-90 triangle has sides x, x\sqrt{3}, and 2x, shortest side opposite the 30^\circ angle. The one skill the sheet cannot supply is mapping: given one side, identify which slot it fills, then scale the other two. Most misses come from assigning x\sqrt{3} to the wrong side, not from forgetting the pattern.
Sine of an angle, cosine of its complement
The two acute angles of a right triangle add to 90^\circ, and the side opposite one is the side adjacent to the other. So \sin A = \cos B whenever A + B = 90^\circ. If a question says \sin(x^\circ) = \cos(y^\circ), the entire content of the question is x + y = 90. No triangle needs drawing, no calculator helps. (Angles here stay in degrees; radians live over in circle questions.)
If \sin(x^\circ) = \cos(38^\circ) and 0 < x < 90, what is the value of x?
Drill the ratios, the families, and the complement identity with a tutor that makes you label the sides out loud.
Practice Right Triangles & Trig →Frequently asked questions
- What trigonometry do you actually need for the SAT?
- Right-triangle trig only: the Pythagorean theorem, the sine, cosine, and tangent ratios, the two special triangles, and the identity that sine of an angle equals cosine of its complement. The unit circle and trig graphs are not tested.
- What are the special right triangles on the SAT?
- 45-45-90, with legs equal and hypotenuse leg-times-root-2, and 30-60-90, with sides in the ratio x, x root 3, and 2x, shortest side opposite the 30 degree angle. Both are printed on the reference sheet, so the skill is mapping the given side to the right slot.
- Why does sin(x) equal cos(90 - x)?
- The two acute angles in a right triangle sum to 90 degrees, and the side opposite one is the side adjacent to the other, so sine for one angle is the same fraction as cosine for its complement. On the SAT, sin(x) = cos(y) translates directly to x + y = 90.
- How do I know which side is the hypotenuse?
- It is the side opposite the right angle, always the longest side. Identify it before using the Pythagorean theorem or any trig ratio, because plugging a leg into the hypotenuse slot is the most common error in this category.