Right Triangle Trig and Circle Equations

Right Triangle Trigonometry (SOH-CAH-TOA)

The SAT tests basic trig ratios in applied contexts — never abstract identities.

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Complementary Angle Relationship

This appears frequently on the SAT:

$\sin(x) = \cos(90° - x)$

Example: If $\sin(35°) = 0.574$, what is $\cos(55°)$?

Since $35° + 55° = 90°$, $\cos(55°) = \sin(35°) = 0.574$.

SAT Trig Application

A 20-foot ladder leans against a wall, making a 65° angle with the ground. How high up the wall does the ladder reach?

The height is the side opposite the 65° angle. The ladder is the hypotenuse.

$\sin(65°) = \frac{h}{20} \Rightarrow h = 20\sin(65°) \approx 20(0.906) \approx 18.1 \text{ feet}$

Circle Equations

The standard form of a circle equation is:

$(x - h)^2 + (y - k)^2 = r^2$

where $(h, k)$ is the center and $r$ is the radius.

SAT Pattern: Completing the square to find center and radius.

What is the center of the circle $x^2 + y^2 - 6x + 4y = 12$?

Group and complete the square:

• $x$ terms: $x^2 - 6x + 9 = (x - 3)^2$ (add 9)
• $y$ terms: $y^2 + 4y + 4 = (y + 2)^2$ (add 4)

Add $9 + 4 = 13$ to both sides: $(x - 3)^2 + (y + 2)^2 = 25$

Center: $(3, -2)$, Radius: $5$.

Arc Length and Sector Area

For a central angle $\theta$ (in degrees):

$\text{Arc length} = \frac{\theta}{360} \times 2\pi r$

$\text{Sector area} = \frac{\theta}{360} \times \pi r^2$

Key insight: These are just the fraction of the full circle corresponding to the central angle. A $90°$ angle gives $1/4$ of the circumference (arc) or area (sector).

Radian Measure

The SAT occasionally uses radians. Key conversions:

• $180° = \pi$ radians
• $360° = 2\pi$ radians

In radians: Arc length $= r\theta$ and Sector area $= \frac{1}{2}r^2\theta$.