SAT Math — sine and cosine of complementary angles

Two lines in the College Board's description of the SAT's right-triangle skill go beyond SOH-CAH-TOA: use similarity to calculate values of sine, cosine and tangent, and solve problems using the relationship between sine and cosine of complementary angles. The second one shows up on the test in a form that looks like algebra rather than trigonometry, an equation such as sin((3k + 10)°) = cos((2k − 5)°) with a request for k, and it is often a typed-answer question with no options to check against. This material is built around that relationship and the reasoning behind it.

The core idea is simple: in a right triangle, the leg opposite one acute angle is the leg adjacent to the other, so the sine of one acute angle is the cosine of its complement. The twelve quiz questions approach it from several directions. Students match cos 58° with sin 32°, read cos B from sin A in a drawn right triangle, decide what must be true when sin x° = cos y° for two acute angles, and solve the algebraic versions: a linear expression in k, a pair of angles with a = 2b, a triangle whose acute angles are given as (a + 14)° and (3a − 4)°, and an equation whose answer is a decimal, 22.5. A ladder leaning against a wall puts the same relation into a physical setting.

Two questions make sure the complement rule is not overused. One asks for cos A when sin A is known, which needs the Pythagorean relation for a single angle, not the complement rule for two. Another has students find tan B from tan A, where the result is a reciprocal rather than an equal value. A question on similar right triangles shows why the ratios do not change when every side is halved, and a drawn 8-15-17 triangle asks for cos(90° − A).

Six questions come with drawings of the triangles or of the ladder, with the given sides or angle expressions marked and the requested value left blank. Every explanation names the slip behind each wrong option: making the angles add to 180 instead of 90, setting the two angles equal, reporting the angle when the variable was asked for, or using the hypotenuse where a tangent needs two legs.

The flashcards cover the cofunction relations, SOH-CAH-TOA, the Pythagorean relation, reciprocal tangents and the fact that similar triangles share their ratios.

The material offers a quiz and flashcards. It is practice written by Zestly and is not affiliated with or endorsed by the College Board.

• Use sin x° = cos(90 − x)° to rewrite and compare trigonometric values
• Solve equations such as sin((3k + 10)°) = cos((2k − 5)°) for an unknown
• Relate sine, cosine and tangent of the two acute angles of a right triangle, including reciprocal tangents
• Distinguish the complement relation from the Pythagorean relation for a single angle
• Use similarity to explain why trigonometric ratios do not depend on triangle size

Practice material written by Zestly, based on the College Board Assessment Framework for the Digital SAT Suite (version 3.01, August 2024), Geometry and Trigonometry domain, skill Right triangles and trigonometry.