SAT Math – Geometry and Trigonometry
Right Triangle Trigonometry
Mastering sine, cosine, tangent, and the Pythagorean theorem
Right triangle trigonometry connects angle measures to side ratios in triangles containing a 90° angle. On the SAT, you'll use sine, cosine, and tangent ratios, apply the Pythagorean theorem, recognize special right triangles (30-60-90 and 45-45-90), and solve for missing sides and angles in real-world contexts.
Success requires memorizing SOH-CAH-TOA, understanding complementary angle relationships, applying the Pythagorean theorem automatically, and recognizing when problems involve right triangles. These techniques aren't just abstract mathematics—they enable calculating heights using shadows, determining distances using angles of elevation, engineering ramp slopes, and any application requiring angle-distance relationships.
Understanding Trigonometric Ratios
SOH-CAH-TOA (The Essential Mnemonic)
Trigonometric ratios relate angles to side lengths in right triangles.
Cosine: \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\) (CAH)
Tangent: \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\) (TOA)
Remember: Always relative to the reference angle
Pythagorean Theorem
Relates the three sides of any right triangle.
a, b: Legs (sides forming right angle)
c: Hypotenuse (longest side, opposite right angle)
Use: Find missing side when two sides known
Special Right Triangles
Two triangle types with predictable side ratios.
30-60-90: Sides in ratio \(1:\sqrt{3}:2\) (short leg : long leg : hypotenuse)
Key: No calculation needed—just apply ratios
Shortcut: Recognize these patterns instantly
Complementary Angle Relationships
In right triangles, the two acute angles are complementary (sum to 90°).
Example: \(\sin(30°) = \cos(60°)\)
Also: \(\tan(\theta) = \frac{1}{\tan(90° - \theta)}\)
Insight: Sine and cosine are complementary functions
Essential Formulas and Values
Common Trigonometric Values
30°: \(\sin(30°) = \frac{1}{2}\), \(\cos(30°) = \frac{\sqrt{3}}{2}\), \(\tan(30°) = \frac{1}{\sqrt{3}}\)
45°: \(\sin(45°) = \frac{\sqrt{2}}{2}\), \(\cos(45°) = \frac{\sqrt{2}}{2}\), \(\tan(45°) = 1\)
60°: \(\sin(60°) = \frac{\sqrt{3}}{2}\), \(\cos(60°) = \frac{1}{2}\), \(\tan(60°) = \sqrt{3}\)
Pythagorean Theorem Applications
\(a^2 + b^2 = c^2\)
Find hypotenuse: \(c = \sqrt{a^2 + b^2}\)
Find leg: \(a = \sqrt{c^2 - b^2}\)
Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
Solving for Sides and Angles
Given angle and side, find side: Use sin, cos, or tan
Given two sides, find angle: Use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹)
Given two sides, find third: Use Pythagorean theorem
Strategy: Identify what you have and what you need
Key Identities
Pythagorean identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\)
Tangent definition: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
Complementary: \(\sin(\theta) = \cos(90° - \theta)\)
Common Pitfalls & Expert Tips
❌ Using wrong side relative to angle
Opposite and adjacent depend on WHICH angle you're using! Redraw and label sides relative to your reference angle.
❌ Calculator in wrong mode (degrees vs radians)
SAT typically uses degrees. Ensure calculator is in DEGREE mode, not radians!
❌ Confusing sin⁻¹ with (sin)⁻¹
sin⁻¹ is inverse sine (arcsin), not 1/sin. Use INV or 2nd function button on calculator.
❌ Forgetting to check special triangles first
If you see 30°, 45°, or 60°, use special triangle ratios—faster than trig functions!
✓ Expert Tip: Draw and label clearly
Mark the right angle, label the reference angle, and identify opposite, adjacent, and hypotenuse. Visual clarity prevents errors!
✓ Expert Tip: Use Pythagorean triples
Recognize 3-4-5, 5-12-13, and their multiples (6-8-10, 9-12-15). Save calculation time!
✓ Expert Tip: SOH-CAH-TOA every time
Don't rely on memory alone. Quickly recite SOH-CAH-TOA to verify which ratio to use!
Fully Worked SAT-Style Examples
In a right triangle, one angle measures 30° and the hypotenuse is 20. What is the length of the side opposite the 30° angle?
Solution:
Identify what we have and need:
Angle = 30°, Hypotenuse = 20
Need: Opposite side
Choose formula (SOH):
\(\sin(30°) = \frac{\text{Opposite}}{20}\)
Solve:
\(\frac{1}{2} = \frac{\text{Opposite}}{20}\)
Opposite = 10
Answer: 10
A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
Solution:
Apply Pythagorean theorem:
\(a^2 + b^2 = c^2\)
\(5^2 + 12^2 = c^2\)
\(25 + 144 = c^2\)
\(169 = c^2\)
\(c = 13\)
Recognition:
This is the 5-12-13 Pythagorean triple!
Could recognize immediately without calculation
Answer: 13
In a 45-45-90 triangle, one leg has length 7. What is the length of the hypotenuse?
Solution:
Recall 45-45-90 ratio:
Leg : Leg : Hypotenuse = \(1:1:\sqrt{2}\)
Apply ratio:
If leg = 7, then hypotenuse = \(7\sqrt{2}\)
Answer: \(7\sqrt{2}\)
In a 30-60-90 triangle, the hypotenuse is 10. What is the length of the shorter leg?
Solution:
Recall 30-60-90 ratio:
Short leg : Long leg : Hypotenuse = \(1:\sqrt{3}:2\)
Set up proportion:
\(\frac{\text{Short leg}}{1} = \frac{10}{2}\)
Short leg = 5
Answer: 5
In a right triangle, one angle is 60° and the adjacent side to this angle is 5. What is the length of the opposite side?
Solution:
Choose formula (TOA):
\(\tan(60°) = \frac{\text{Opposite}}{5}\)
Solve:
\(\sqrt{3} = \frac{\text{Opposite}}{5}\)
Opposite = \(5\sqrt{3}\)
Answer: \(5\sqrt{3}\)
A right triangle has a hypotenuse of 10 and one leg of 6. What is the length of the other leg?
Solution:
Use Pythagorean theorem:
\(6^2 + b^2 = 10^2\)
\(36 + b^2 = 100\)
\(b^2 = 64\)
\(b = 8\)
Recognition:
This is a 6-8-10 triangle (multiple of 3-4-5)
Answer: 8
A ladder 15 feet long leans against a wall at a 60° angle to the ground. How high up the wall does the ladder reach?
Solution:
Identify components:
Ladder = hypotenuse = 15 ft
Angle with ground = 60°
Height = opposite side
Use sine (SOH):
\(\sin(60°) = \frac{\text{Height}}{15}\)
\(\frac{\sqrt{3}}{2} = \frac{\text{Height}}{15}\)
Height = \(\frac{15\sqrt{3}}{2}\) ft
Answer: \(\frac{15\sqrt{3}}{2}\) ft (approximately 13.0 ft)
Quick Reference Chart
SOH-CAH-TOA
sin = Opp/Hyp
cos = Adj/Hyp
tan = Opp/Adj
Special Triangles
45-45-90: \(1:1:\sqrt{2}\)
30-60-90: \(1:\sqrt{3}:2\)
Right Triangle Trigonometry: Connecting Angles and Distances
Right triangle trigonometry provides the mathematical bridge between angles and distances, enabling indirect measurement and spatial reasoning essential across engineering, navigation, physics, and architecture. The SAT tests these skills because they represent fundamental quantitative thinking—understanding that consistent angle-side ratios in right triangles (sine, cosine, tangent) allow calculating inaccessible dimensions from measurable ones. The mnemonic SOH-CAH-TOA encodes these relationships: sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent—simple ratios that unlock countless applications from surveying building heights using angle measurements to calculating distances across rivers using triangulation. The Pythagorean theorem \(a^2 + b^2 = c^2\) provides the fundamental relationship among right triangle sides, while special triangles (45-45-90 with ratio \(1:1:\sqrt{2}\) and 30-60-90 with ratio \(1:\sqrt{3}:2\)) offer instant solutions without calculation when recognized. Understanding complementary angle relationships—that sine of an angle equals cosine of its complement—reveals the deep symmetry in trigonometric functions. These techniques transcend classroom exercises, empowering you to calculate shadow lengths for solar panel installation, determine ramp slopes for accessibility requirements, analyze forces in physics problems involving inclined planes, navigate using bearings and distances, and understand how GPS systems triangulate positions using angle measurements from multiple satellites. The discipline of identifying which sides are opposite, adjacent, and hypotenuse relative to a reference angle develops spatial reasoning crucial for interpreting technical diagrams, understanding three-dimensional structures from two-dimensional projections, and translating between angular and linear measurements. Every tangent calculation for slope, every sine calculation for component forces, every Pythagorean application for diagonal distances represents practical problem-solving connecting geometry to the physical world—skills fundamental to STEM fields and spatial literacy in everyday contexts from construction to sports to technology.