Home / Courses / SAT Math / Unit 8

SAT Math Unit 8: Right triangles, trigonometry & circles

Right triangles and circles round out the SAT geometry domain. Expect SOH-CAH-TOA, special triangles and circle equations.

Covers: Pythagorean theorem, special right triangles, sine, cosine and tangent, circle equations, arcs and sectors

Study guide

The Pythagorean theorem, a² + b² = c², relates the legs and hypotenuse of a right triangle. Common triples like 3-4-5, 5-12-13 and their multiples save time. Two special right triangles appear constantly: a 45-45-90 triangle has sides x, x, x√2, and a 30-60-90 triangle has sides x, x√3, 2x, with the shortest side opposite the 30° angle.

Right-triangle trigonometry uses SOH-CAH-TOA: sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, tangent is opposite/adjacent. In a right triangle the two acute angles are complementary, so sin x° = cos(90 − x)°. Radians measure angles by arc length: 180° = π radians.

A circle with center (h, k) and radius r has equation (x − h)² + (y − k)² = r². If an equation is in expanded form, complete the square on x and on y to find the center and radius.

Arcs and sectors are fractions of the whole circle. Arc length is (central angle/360°)·2πr and sector area is (central angle/360°)·πr². An inscribed angle is half the central angle that cuts off the same arc, and a tangent line is perpendicular to the radius at the point of contact.

To convert degrees to radians multiply by π/180, and to go back multiply by 180/π. With θ in radians, arc length is s = rθ and sector area is ½r²θ. A trig ratio depends only on the angle, not the size of the triangle, so corresponding angles in similar triangles have equal sine, cosine and tangent values. This lets you carry a ratio from one triangle to another.

On the unit circle, an angle's terminal point is (cos θ, sin θ), so cosine is negative in Quadrants II and III and sine is negative in Quadrants III and IV. The values for an angle equal those of its reference angle, with the sign set by the quadrant.

Key ideas

• a² + b² = c² for right triangles.
• 45-45-90: sides x, x, x√2. 30-60-90: x, x√3, 2x.
• sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
• sin x° = cos(90 − x)°.
• Circle: (x − h)² + (y − k)² = r². Complete the square to find the center and radius.

Formulas and rules

Common mistakes

• Reading r² as the radius in a circle equation.
• Mixing up sign conventions for the center (x − 2 means h = +2).

Words to know

Sides of a 45-45-90 triangle

x, x, x√2

Sides of a 30-60-90 triangle

x, x√3, 2x

Cofunction identity

sin θ = cos(90° − θ).

Circle equation (standard form)

(x − h)² + (y − k)² = r²: center (h, k), radius r.

Arc length formula

(central angle/360)·2πr

Pythagorean theorem

a² + b² = c² for a right triangle with hypotenuse c.

SOH-CAH-TOA

sin = opp/hyp, cos = adj/hyp, tan = opp/adj.

Radian

An angle measure where 180° = π radians.

Sector area

(central angle/360°)·πr²

Inscribed angle

Half the central angle that intercepts the same arc.

Tangent to a circle

Perpendicular to the radius at the point of contact.

Trig ratios in right triangles

sin = opp/hyp, cos = adj/hyp, tan = opp/adj, relative to an acute angle.

Sample practice questions

297 more questions like these, from level 1 to 5, are in the practice set.