Worked Examples: SAT Strategy
Example 1: Easy (Pythagorean Triples & SOH CAH TOA)
Question: In right triangle $ABC$, the right angle is at $B$. If $AC = 13$ and $AB = 5$, what is the value of $\tan(C)$?
SAT Strategy: Recognize common Pythagorean triples to save time.
- The triangle has a hypotenuse of $13$ ($AC$) and a leg of $5$ ($AB$). This is a classic $5-12-13$ right triangle. Therefore, the missing leg $BC = 12$.
- We need $\tan(C)$. Tangent is Opposite over Adjacent (TOA).
- From angle $C$, the opposite leg is $AB$ ($5$) and the adjacent leg is $BC$ ($12$).
- $\tan(C) = \frac{5}{12}$.
- Answer: $\frac{5}{12}$.
Example 2: Medium (Complementary Angle Theorem)
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 use your calculator to find inverse sine ($\sin^{-1}$)! The SAT tests a specific rule here: The sine of an angle is always exactly equal to the cosine of its complement.
- In a right triangle, the two non-right angles add up to $90^\circ$. They are complementary.
- The side that is "opposite" to angle $x$ is the exact same side that is "adjacent" to the other angle ($90 - x$).
- Therefore, $\sin(x) = \cos(90 - x)$.
- If $\sin(x^\circ) = \frac{4}{5}$, then $\cos(90^\circ - x^\circ)$ is also $\frac{4}{5}$.
- Answer: $\frac{4}{5}$ (or $0.8$).
Example 3: Hard (Special Right Triangles)
Question: A square has a diagonal of $8\sqrt{2}$ inches. What is the perimeter of the square, in inches?
SAT Strategy: Use the $45^\circ-45^\circ-90^\circ$ special right triangle formulas provided on the SAT reference sheet. A square split by its diagonal creates two $45^\circ-45^\circ-90^\circ$ triangles.
- The reference sheet states that the hypotenuse of a $45^\circ-45^\circ-90^\circ$ triangle is $x\sqrt{2}$, where $x$ is the side length.
- The diagonal is the hypotenuse: $x\sqrt{2} = 8\sqrt{2}$.
- Therefore, the side length $x = 8$.
- A square has 4 equal sides. The perimeter is $4 \times 8 = 32$.
- Answer: 32.
Practice Questions
Level 1: Easy (Foundational SAT)
- In right triangle $PQR$, the right angle is at $Q$. If $\sin(P) = \frac{3}{5}$, what is the value of $\cos(R)$? (Grid-in)
- The lengths of the two legs of a right triangle are $6$ cm and $8$ cm. What is the length of the hypotenuse, in cm?
- A) $10$
- B) $12$
- C) $14$
- D) $100$
- In a right triangle, $\sin(30^\circ) = 0.5$. What is the value of $\cos(60^\circ)$? (Grid-in)
- In right triangle $ABC$, angle $C$ is $90^\circ$. If $AC = 7$ and $BC = 24$, what is the length of $AB$?
- A) $17$
- B) $25$
- C) $31$
- D) $625$
- Triangle $DEF$ is a right triangle with the right angle at $E$. If $\tan(D) = \frac{15}{8}$, what is the value of $\frac{EF}{DE}$?
- A) $\frac{8}{15}$
- B) $\frac{15}{17}$
- C) $\frac{15}{8}$
- D) $\frac{17}{8}$
Level 2: Medium (Standard SAT)
- In a right triangle, $\cos(x^\circ) = \frac{2}{3}$. What is the value of $\sin(90^\circ - x^\circ)$? (Grid-in)
- The diagonal of a square is $10$ units long. What is the area of the square, in square units?
- A) $25$
- B) $50$
- C) $75$
- D) $100$
- In triangle $XYZ$, angle $Y$ is a right angle. If $\sin(X) = \frac{12}{13}$, what is the value of $\tan(Z)$?
- A) $\frac{5}{12}$
- B) $\frac{5}{13}$
- C) $\frac{12}{5}$
- D) $\frac{13}{5}$
- An equilateral triangle has a side length of $6$ inches. What is the height of the triangle, in inches? (Hint: Use the $30^\circ-60^\circ-90^\circ$ rule).
- A) $3$
- B) $3\sqrt{2}$
- C) $3\sqrt{3}$
- D) $6\sqrt{3}$
- A right triangle has one leg with length $x$ and a hypotenuse with length $2x$. What is the measure of the angle opposite the leg of length $x$? (Grid-in)
Level 3: Hard (Advanced SAT)
- In a right triangle, the two acute angles are $A$ and $B$. If $\sin(A) = \frac{3}{7}$, what is the value of $\sin^2(A) + \cos^2(A)$?
- A) $\frac{9}{49}$
- B) $\frac{40}{49}$
- C) $1$
- D) It cannot be determined from the given information.
- If $\sin(3x - 15)^\circ = \cos(2x + 10)^\circ$, what is the value of $x$? (Grid-in)
- A 20-foot ladder is leaning against a vertical wall. The angle the ladder makes with the ground is $60^\circ$. How far is the base of the ladder from the wall, in feet? (Grid-in)
- Triangle $ABC$ is similar to triangle $DEF$, where angle $A$ corresponds to angle $D$ and angle $C$ corresponds to angle $F$. Both $C$ and $F$ are right angles. If $\sin(A) = 0.28$, what is the value of $\sin(D)$?
- A) $0.28$
- B) $0.72$
- C) $0.96$
- D) It cannot be determined from the given information.
- In a right triangle, the tangent of one of the acute angles is $\frac{x}{y}$. What is the tangent of the other acute angle?
- A) $\frac{x}{y}$
- B) $\frac{y}{x}$
- C) $1$
- D) $\frac{x^2}{y^2}$
Level 4: Very Hard (Slightly beyond SAT)
- In right triangle $RST$, angle $S$ is a right angle. Point $U$ lies on the hypotenuse $RT$ such that line segment $SU$ is perpendicular to $RT$. If $\sin(R) = \frac{4}{5}$, what is the value of $\cos(\angle TSU)$?
- A) $\frac{3}{5}$
- B) $\frac{4}{5}$
- C) $\frac{3}{4}$
- D) $\frac{4}{3}$
- A square is inscribed perfectly inside a circle with a radius of $8$ units. What is the perimeter of the square?
- A) $32$
- B) $32\sqrt{2}$
- C) $64$
- D) $64\sqrt{2}$
- In right triangle $ABC$, the right angle is $C$. If $\sin(A) = \frac{m}{n}$, which of the following expressions represents $\tan(B)$?
- A) $\frac{n}{m}$
- B) $\frac{m}{\sqrt{n^2 - m^2}}$
- C) $\frac{\sqrt{n^2 - m^2}}{m}$
- D) $\frac{\sqrt{n^2 - m^2}}{n}$
- The measures of two acute angles are $x^\circ$ and $y^\circ$, such that $\sin(x^\circ) = \cos(y^\circ)$. If $x = 3a - 14$ and $y = 5a + 16$, what is the value of $x$? (Grid-in)
- Two ships leave a port at the exact same time. Ship A travels due North for $3k$ miles. Ship B travels due East for $4k$ miles. If the direct distance between the two ships is $60$ miles, what is the value of $k$? (Grid-in)
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