Worked Examples: SAT Strategy
Example 1: Easy (The Complementary Angle Shortcut)
Question: In a right triangle, one angle measures $x^\circ$, where $\sin(x^\circ) = \frac{4}{5}$. What is $\cos(90^\circ - x^\circ)$?
SAT Strategy: Do not draw the triangle or calculate the angle using $\arcsin$! The SAT tests the specific trigonometric identity that the sine of an angle is always exactly equal to the cosine of its complement.
- Recognize the structure: $\cos(90^\circ - x^\circ)$ is the complement of $\sin(x^\circ)$.
- Apply the identity: $\sin(x^\circ) = \cos(90^\circ - x^\circ)$.
- Since $\sin(x^\circ) = \frac{4}{5}$, then $\cos(90^\circ - x^\circ)$ must also equal $\frac{4}{5}$.
- Answer: $\frac{4}{5}$ (or $0.8$).
Example 2: Medium (Completing the Square for Circles)
Question: The equation of a circle in the $xy$-plane is $x^2 + y^2 - 8x + 10y = 8$. What is the radius of the circle?
SAT Strategy: The SAT rarely gives you the circle equation in the easy $(x-h)^2 + (y-k)^2 = r^2$ format. You must "complete the square" for both $x$ and $y$. Take half the middle coefficient, square it, and add it to BOTH sides.
- Group the terms: $(x^2 - 8x) + (y^2 + 10y) = 8$.
- Complete the $x$ square: Half of $-8$ is $-4$. Squared is $16$. Add $16$ to both sides.
- Complete the $y$ square: Half of $10$ is $5$. Squared is $25$. Add $25$ to both sides.
- New equation: $(x^2 - 8x + 16) + (y^2 + 10y + 25) = 8 + 16 + 25$.
- Factor and simplify: $(x - 4)^2 + (y + 5)^2 = 49$.
- Since $r^2 = 49$, the radius is $\sqrt{49} = 7$.
- Answer: $7$.
Example 3: Hard (Proportional Arcs and Sectors)
Question: A circle has a total area of $120\pi$. A sector of the circle has an area of $24\pi$. What is the measure of the central angle of this sector, in radians?
SAT Strategy: Do not use complex IB sector area formulas if you can use simple proportions. The ratio of the sector area to the total area is identical to the ratio of the central angle to the total angle ($2\pi$ radians).
- Set up the ratio of the areas: $\frac{\text{Sector Area}}{\text{Total Area}} = \frac{24\pi}{120\pi} = \frac{1}{5}$.
- This means the sector represents exactly $\frac{1}{5}$ of the entire circle.
- The total angle of a circle in radians is $2\pi$.
- Multiply the fraction by the total angle: $\frac{1}{5} \times 2\pi = \frac{2\pi}{5}$.
- Answer: $\frac{2\pi}{5}$.
Practice Questions
Level 1: Easy (Foundational SAT)
- In a right triangle $ABC$, the sine of acute angle $A$ is $\frac{7}{25}$. What is the cosine of acute angle $B$?
- A) $\frac{7}{25}$
- B) $\frac{24}{25}$
- C) $\frac{25}{7}$
- D) $\frac{7}{24}$
- A circle in the $xy$-plane has the equation $(x + 3)^2 + (y - 5)^2 = 64$. What is the center of the circle?
- A) $(3, -5)$
- B) $(-3, 5)$
- C) $(9, 25)$
- D) $(-3, -5)$
- What is the radius of the circle given by the equation $(x - 1)^2 + y^2 = 144$? (Grid-in)
- In right triangle $PQR$, angle $Q$ measures $90^\circ$. If $PQ = 6$ and $QR = 8$, what is the length of the hypotenuse $PR$? (Grid-in)
- A central angle in a circle measures $\frac{\pi}{3}$ radians. What is the measure of this angle in degrees?
- A) $30^\circ$
- B) $60^\circ$
- C) $90^\circ$
- D) $120^\circ$
Level 2: Medium (Standard SAT)
- In a right triangle, $\sin(k^\circ) = \frac{5}{13}$. If $\cos(m^\circ) = \frac{5}{13}$ and $k$ and $m$ are acute angles, what is the value of $k + m$? (Grid-in)
- The circle $x^2 + y^2 = 25$ is intersected by the line $y = 4$ at two points. What is the length of the straight-line segment connecting these two points of intersection?
- A) $3$
- B) $6$
- C) $8$
- D) $10$
- A circle has a circumference of $18\pi$. What is the area of a sector of this circle with a central angle of $40^\circ$?
- A) $2\pi$
- B) $4\pi$
- C) $9\pi$
- D) $18\pi$
- The equation of a circle is $x^2 + y^2 + 10x - 4y = 7$. What is the radius of this circle? (Grid-in)
- In a right triangle, the tangent of one of the acute angles is $\frac{3}{4}$. What is the sine of that same angle?
- A) $\frac{3}{7}$
- B) $\frac{3}{5}$
- C) $\frac{4}{5}$
- D) $\frac{5}{3}$
Level 3: Hard (Advanced SAT)
- The endpoints of a diameter of a circle are $(-4, 2)$ and $(6, -2)$. Which of the following is the equation of the circle?
- A) $(x - 1)^2 + y^2 = 29$
- B) $(x - 1)^2 + y^2 = 116$
- C) $(x + 1)^2 + y^2 = 29$
- D) $(x + 1)^2 + y^2 = 116$
- An arc of a circle has a length of $5\pi$ and the radius of the circle is $12$. What is the measure of the central angle that subtends this arc, in radians? (Grid-in)
- The equation $x^2 + y^2 - 12x + 16y = -k$ represents a circle in the $xy$-plane. If the radius of the circle is $8$, what is the value of the constant $k$? (Grid-in)
- In $\triangle ABC$, $\angle B = 90^\circ$ and $BD$ is the altitude to the hypotenuse $AC$. If $AD = 4$ and $DC = 9$, what is the length of $BD$?
- A) $6$
- B) $6.5$
- C) $13$
- D) $36$
- A wheel with a radius of $15$ inches rolls forward without slipping. If the wheel completes exactly $4.5$ revolutions, how many inches did the wheel travel?
- A) $67.5\pi$
- B) $135\pi$
- C) $225\pi$
- D) $1012.5\pi$
Level 4: Very Hard (Slightly beyond SAT)
- A circle in the $xy$-plane has a center at $(3, 5)$. The line $y = -\frac{1}{2}x + 14$ is tangent to the circle at point $P$. What are the coordinates of point $P$?
- A) $(6, 11)$
- B) $(5, 11.5)$
- C) $(8, 10)$
- D) $(4, 12)$
- A square is perfectly inscribed within a circle. If the area of the square is $100$ square units, what is the exact area of the circle?
- A) $25\pi$
- B) $50\pi$
- C) $100\pi$
- D) $200\pi$
- If $\sin(3x - 15)^\circ = \cos(2x + 20)^\circ$, and $x$ is a positive constant, what is one possible value of $x$? (Grid-in)
- A right triangle has side lengths of $x$, $x + 7$, and $x + 14$. What is the perimeter of the triangle? (Grid-in)
- A circle has the equation $2x^2 + 2y^2 - 20x + 24y = -10$. What is the area of this circle?
- A) $56\pi$
- B) $112\pi$
- C) $224\pi$
- D) $448\pi$
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