Mastering Trigonometry on the SAT

Trigonometry is a key component of the SAT math section. This guide provides a comprehensive approach to mastering trigonometric concepts for the SAT.

Trigonometry deals with relationships between angles and sides of triangles. It involves using right triangles to find missing angle measures and side lengths.

Study priority: Trig questions usually appear 1-2 times per exam. If you only have a few weeks to study and are not familiar with trig, solidify algebra and nonlinear functions first as those appear much more often.

SOHCAHTOA and the Pythagorean Theorem

SOHCAHTOA is the mnemonic for trig ratios: SOH = Sine = Opposite/Hypotenuse, CAH = Cosine = Adjacent/Hypotenuse, TOA = Tangent = Opposite/Adjacent.

The Pythagorean theorem (a^2 + b^2 = c^2) defines the relationship between sides of a right triangle. Combining SOHCAHTOA with the Pythagorean theorem lets you solve a wide range of problems.

Example: In a 30-degree right triangle with hypotenuse 2: opposite = 1 (sin 30 = 1/2), adjacent = sqrt(3). So sin 30 = 1/2, cos 30 = sqrt(3)/2, tan 30 = 1/sqrt(3).

Reciprocal Trigonometric Functions

Secant (sec) = 1/cos, Cosecant (csc) = 1/sin, Cotangent (cot) = 1/tan. These provide alternative ways to solve problems.

Example: If sin 45 = sqrt(2)/2, then csc 45 = 1/(sqrt(2)/2) = 2/sqrt(2) = sqrt(2).

Trigonometric Identities

Pythagorean identities: sin^2(x) + cos^2(x) = 1, 1 + tan^2(x) = sec^2(x), 1 + cot^2(x) = csc^2(x).

Quotient identities: tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x).

Example: sin(45) * sec(45) = (sqrt(2)/2) * sqrt(2) = 2/2 = 1.

Radians

One full revolution = 2pi radians = 360 degrees. To convert: Degrees = Radians x 180/pi. Radians = Degrees x pi/180.

Example: 45 degrees = 45 x pi/180 = pi/4 radians.

The Unit Circle

A circle with radius 1 centered at the origin. Equation: x^2 + y^2 = 1. For any angle theta: cos(theta) = x-coordinate, sin(theta) = y-coordinate, tan(theta) = y/x.

Example: At pi/3: coordinates are (cos(pi/3), sin(pi/3)) = (1/2, sqrt(3)/2).

Graphing Trigonometric Functions

Sine and cosine are periodic with period 2pi. Tangent has period pi. The general form y = A*sin(Bx + C) + D has amplitude A, period 2pi/B, horizontal shift C, vertical shift D.

Example: Period of y = sin(3x): period = 2pi/3.

Special Right Triangles

30-60-90 Triangle

Side ratios: 1 : sqrt(3) : 2. Shorter leg = a, longer leg = a*sqrt(3), hypotenuse = 2a. If hypotenuse is 10: shorter leg = 5, longer leg = 5*sqrt(3).

45-45-90 Triangle

Side ratios: 1 : 1 : sqrt(2). Legs = a, hypotenuse = a*sqrt(2). If hypotenuse is 10: legs = 10/sqrt(2) = 5*sqrt(2).

These triangles frequently appear on the SAT and allow quick calculation without needing the Pythagorean theorem or trig functions.

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Frequently Asked Questions

A mnemonic for trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

The 30-60-90 (sides 1:sqrt(3):2) and 45-45-90 (sides 1:1:sqrt(2)). These have fixed ratios for quick side calculations.

Degrees to radians: multiply by pi/180. Radians to degrees: multiply by 180/pi. Example: 45 degrees = pi/4 radians.

Usually 1-2 per exam. Prioritize algebra first if study time is limited.