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Trigonometry

Definition

Trigonometry is the branch of mathematics that studies relationships between side lengths and angles of triangles. It primarily involves trigonometric functions (sine, cosine, tangent, and their reciprocals) which relate angles to ratios of side lengths in right-angled triangles and are extended to all angles using the unit circle.

Prerequisites

A solid understanding of these topics is crucial:

  1. Coordinate Geometry (Coordinate plane, distance)
  2. Pythagorean Theorem
  3. Angle Relationships (Degrees, basic angles)
  4. Basic Functions (Function notation, domain, range)
  5. Circle Theorems (Optional, but helpful for unit circle understanding)
  6. Algebraic Manipulation

Learning Objectives

After mastering this topic, you should be able to:

  1. Define the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) using right triangle ratios (SOH CAH TOA) and the unit circle (x, y coordinates).
  2. Evaluate trigonometric functions for common angles (0°, 30°, 45°, 60°, 90°, and their multiples) using special right triangles and the unit circle.
  3. Understand radian measure and convert between degrees and radians.
  4. Graph the basic sine, cosine, and tangent functions and identify their properties (amplitude, period, phase shift, vertical shift).
  5. Apply trigonometric functions to solve right triangles (find missing sides or angles).
  6. Use the Law of Sines and the Law of Cosines to solve non-right (oblique) triangles.
  7. Understand and apply basic trigonometric identities (Reciprocal, Quotient, Pythagorean).
  8. Solve basic trigonometric equations.

Key Concepts

Right Triangle Trigonometry (SOH CAH TOA)

  • For an acute angle θ in a right triangle:
    • sin(θ) = Opposite / Hypotenuse
    • cos(θ) = Adjacent / Hypotenuse
    • tan(θ) = Opposite / Adjacent
  • Reciprocal Functions: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ)

Unit Circle Trigonometry

  • A circle centered at the origin with radius 1.
  • For an angle θ in standard position (vertex at origin, initial side on positive x-axis), the terminal side intersects the unit circle at point (x, y).
  • cos(θ) = x
  • sin(θ) = y
  • tan(θ) = y / x
  • Allows definition for any angle, including those > 90° or negative.

Radian Measure

  • An alternative unit for measuring angles (1 radian is the angle subtended when the arc length equals the radius).
  • Conversion: π radians = 180°

Graphs of Trigonometric Functions

  • Periodic (repeat in cycles).
  • Key properties: Amplitude (height from midline), Period (length of one cycle), Phase Shift (horizontal shift), Vertical Shift.

Solving Triangles

  • Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) (Used when given AAS, ASA, SSA - ambiguous case).
  • Law of Cosines: c² = a² + b² - 2ab cos(C) (Used when given SAS, SSS).

Trigonometric Identities

  • Reciprocal: cscθ=1/sinθ, secθ=1/cosθ, cotθ=1/tanθ
  • Quotient: tanθ=sinθ/cosθ, cotθ=cosθ/sinθ
  • Pythagorean: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ

Examples

Example 1: Right Triangle Ratios (High School)

Problem: In a right triangle, the side opposite angle A is 3, the adjacent side is 4, and the hypotenuse is 5. Find sin(A), cos(A), and tan(A). Solution: * sin(A) = Opposite/Hypotenuse = 3/5 * cos(A) = Adjacent/Hypotenuse = 4/5 * tan(A) = Opposite/Adjacent = 3/4

Example 2: Unit Circle Evaluation (High School)

Problem: Find the exact value of cos(150°). Solution: 150° is in Quadrant II. The reference angle is 180° - 150° = 30°. The coordinates on the unit circle for 30° are (√3/2, 1/2). In Quadrant II, x is negative, y is positive. So, the coordinates for 150° are (-√3/2, 1/2). Since cos(θ) = x, cos(150°) = -√3/2.

Example 3: Radian Conversion (High School)

Problem: Convert 120° to radians. Solution: 120° × (π radians / 180°) = 120π / 180 radians = 2π/3 radians.

Example 4: Law of Sines (High School/Precalculus)

Problem: In triangle ABC, angle A = 40°, angle B = 60°, side a = 10. Find side b. Solution: Use Law of Sines: a/sin(A) = b/sin(B) 10 / sin(40°) = b / sin(60°) b = 10 * sin(60°) / sin(40°) ≈ 10 * (0.866) / (0.643) ≈ 13.47

Common Misconceptions

  1. Calculator Mode: Using degrees mode when calculations require radians, or vice versa.
  2. SOH CAH TOA Errors: Mixing up opposite, adjacent, and hypotenuse.
  3. Unit Circle Signs: Forgetting the correct sign (+/-) for sin, cos, tan in different quadrants.
  4. Law of Sines Ambiguous Case (SSA): Forgetting that SSA can lead to zero, one, or two possible triangles.
  5. Identity Errors: Incorrectly applying or remembering trigonometric identities.

Applications in SAT

Trigonometry appears on the SAT, often blending with geometry and algebra:

  1. Right Triangle Trig: Solving for sides/angles using SOH CAH TOA.
  2. Unit Circle: Understanding radian measure and coordinates on the unit circle.
  3. Trigonometric Functions: Basic properties and values for common angles.
  4. Identities: Using Pythagorean identity (sin²θ + cos²θ = 1) is common.
  5. Law of Sines/Cosines: Less common, but may appear in harder problems.

Advanced Connections

Trigonometry is essential for:

  1. Complex Numbers - Polar form (r(cosθ + i sinθ)).
  2. Vectors: Representing vectors in terms of magnitude and direction (angle).
  3. Calculus: Derivatives and integrals of trigonometric functions, trigonometric substitution.
  4. Physics: Analyzing waves, oscillations, forces, projectile motion.
  5. Engineering: Signal processing, structural analysis, circuit analysis.

Practice Problems

  1. Basic: If cos(θ) = 5/13 and θ is in Quadrant IV, find sin(θ).
  2. Intermediate: Convert 5π/6 radians to degrees.
  3. Advanced: Solve the triangle ABC if angle C = 90°, angle A = 35°, and side c = 12.
  4. SAT-Level: In a right triangle, one angle measures x such that sin(x) = 4/5. What is cos(x)?