Overview
SAT trigonometry is limited to right triangle trigonometry — no law of sines or law of cosines is required. The three primary ratios (sine, cosine, tangent) are defined relative to a chosen angle in a right triangle using the SOH-CAH-TOA mnemonic. The Pythagorean identity sin²θ + cos²θ = 1 and the co-function identity sin(θ) = cos(90° − θ) both appear directly on the SAT. Special angle values (30°, 45°, 60°) must be known from memory. Trig questions appear approximately 1–2 times per test.
Key Points
1. SOH-CAH-TOA Definitions
All three ratios are defined relative to a specific angle θ inside a right triangle:
sin θ = Opposite / Hypotenuse (SOH)
cos θ = Adjacent / Hypotenuse (CAH)
tan θ = Opposite / Adjacent (TOA)
Labeling sides relative to θ:
- Hypotenuse: side opposite the right angle (always the longest side)
- Opposite: side directly across from θ
- Adjacent: side next to θ (not the hypotenuse)
The labels “opposite” and “adjacent” switch when you change which angle you’re considering. Always identify θ first before labeling.
2. Reciprocal Trigonometric Functions
csc θ = 1 / sin θ = Hypotenuse / Opposite
sec θ = 1 / cos θ = Hypotenuse / Adjacent
cot θ = 1 / tan θ = Adjacent / Opposite
Memory aid for reciprocals: cosecant goes with sine (not cosine), secant goes with cosine (not sine) — the co- prefix pairs with the opposite function.
3. The Pythagorean Identity
sin²θ + cos²θ = 1
This identity follows directly from the Pythagorean theorem applied to a unit circle. On the SAT it is used to find sin θ when cos θ is given (or vice versa), without needing to know the angle itself.
Example: If cos θ = 3/5, find sin θ.
sin²θ = 1 − cos²θ = 1 − 9/25 = 16/25
sin θ = 4/5 (positive if θ is in the first quadrant)
4. Co-Function Identities
In a right triangle, the two non-right angles are complementary (sum to 90°):
sin(θ) = cos(90° − θ)
cos(θ) = sin(90° − θ)
tan(θ) = cot(90° − θ)
SAT application: A question may state that sin(x°) = cos(y°) and ask for x + y. Since this means y = 90 − x, the answer is x + y = 90.
5. Special Angle Values
These values must be memorized:
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 (= 1/√3) |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Memory pattern:
- sin increases 0 → 1 as angle increases 0° → 90°
- cos decreases 1 → 0 as angle increases 0° → 90°
- At 45°: sin = cos (the triangle is isosceles)
6. Step-by-Step Method for Right Triangle Problems
- Draw and label the right triangle.
- Identify the reference angle θ (not the right angle).
- Label the three sides: hypotenuse (opposite right angle), opposite (across from θ), adjacent (next to θ).
- Determine which two sides are involved: given and unknown.
- Choose the trig ratio that connects those two sides (sin, cos, or tan).
- Write and solve the equation. Use algebra to isolate the unknown.
- To find an angle given two sides, use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹) on a calculator.
Pitfalls and Common Mistakes
Pitfall 1: Calculator mode set to radians when the problem uses degrees This is the most common calculation error in trig. sin(30°) = 0.5 in degree mode but sin(30 radians) ≈ −0.988 in radian mode. The wrong mode gives a completely wrong numerical answer. Fix: Before every trig calculation, verify your calculator is in the correct mode (DEG or RAD) by checking sin(30) = 0.5 (degree mode).
Pitfall 2: Mislabeling opposite and adjacent The “opposite” side is the side across from angle θ, and “adjacent” is the side next to θ. These labels depend entirely on which angle θ you choose. Labeling relative to the wrong angle flips sin and cos. Fix: Circle or mark angle θ on the diagram first. Then label the three sides in order: hypotenuse (across from 90°), opposite (across from θ), adjacent (remaining side).
Pitfall 3: Applying SOH-CAH-TOA to non-right triangles These ratios are only defined for right triangles. If the triangle does not have a 90° angle, SOH-CAH-TOA cannot be directly applied. Fix: Check for the right angle symbol before using trig ratios. If none exists, check whether the problem can be broken into right triangles.
Pitfall 4: Confusing secant and cosecant sec = 1/cos and csc = 1/sin — the prefixes are switched relative to what students expect. Writing sec = 1/sin (or csc = 1/cos) produces wrong answers. Fix: Remember that “cosecant” has the “co-” prefix but goes with sine (the function without “co-”). Alternatively: sec θ = H/A (like cosine but flipped); csc θ = H/O (like sine but flipped).
Related Entries
- Radians_Arc_Measure
- Right_Triangles_Pythagorean
- Similar_Triangles
- Properties_Circles
- Lines_Angles_Parallel
Quick Reference Card
| Ratio | Definition | Reciprocal |
|---|---|---|
| sin θ | O/H | csc θ = H/O |
| cos θ | A/H | sec θ = H/A |
| tan θ | O/A | cot θ = A/O |
| Identity | Formula |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1 |
| Co-function | sin(θ) = cos(90°−θ); cos(θ) = sin(90°−θ) |
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Top trap: calculator in wrong mode (degrees vs. radians)