11.2 Trigonometric Functions and the Unit Circle

Trigonometric functions and the unit circle are key to understanding periodic behavior in math. They help us model real-world cycles and rotations, from sound waves to planetary orbits.

The unit circle connects angle measures to x and y coordinates, defining sine, cosine, and other trig functions. This foundation lets us solve complex problems and analyze cyclic patterns in various fields.

Trigonometric Functions on the Unit Circle

Defining Trigonometric Functions

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• The unit circle is a circle with a radius of 1 centered at the origin (0,0) on a coordinate plane
• An angle θ is formed by the positive x-axis and the line from the origin to a point (x,y) on the unit circle
• The x-coordinate of the point where the terminal side of an angle θ intersects the unit circle equals cos θ
  • In a right triangle, the cosine is the ratio of the adjacent side to the hypotenuse
• The y-coordinate of the point where the terminal side of an angle θ intersects the unit circle equals sin θ
  • In a right triangle, the sine is the ratio of the opposite side to the hypotenuse

Reciprocal Trigonometric Functions

• The tangent (tan θ) is the ratio of the y-coordinate to the x-coordinate on the unit circle
  • In a right triangle, the tangent is the ratio of the opposite side to the adjacent side
• The secant (sec θ), cosecant (csc θ), and cotangent (cot θ) are the reciprocals of cosine, sine, and tangent respectively
  • They can also be defined in terms of the unit circle and right triangle ratios:
    • sec θ = 1/cos θ = hypotenuse/adjacent
    • csc θ = 1/sin θ = hypotenuse/opposite
    • cot θ = 1/tan θ = adjacent/opposite

Evaluating Trigonometric Functions

Using the Unit Circle

• To find the value of a trigonometric function for an angle θ, locate the point (x,y) where the terminal side of the angle intersects the unit circle
  • The x-coordinate is cos θ and the y-coordinate is sin θ
• When the terminal side of an angle lies on an axis, one or both of the coordinates will be 0
  • This results in trigonometric function values of 0, 1, or undefined

Quadrant Rules

• For angles in the first quadrant (0° ≤ θ ≤ 90°), all trigonometric functions are positive
  • The x and y coordinates are both positive
• For angles in the second quadrant (90° < θ ≤ 180°), only sin θ and csc θ are positive
  • The x-coordinate is negative while the y-coordinate is positive
• For angles in the third quadrant (180° < θ ≤ 270°), only tan θ and cot θ are positive
  • Both the x and y coordinates are negative
• For angles in the fourth quadrant (270° < θ < 360°), only cos θ and sec θ are positive
  • The x-coordinate is positive while the y-coordinate is negative

Periodicity of Trigonometric Functions

Periodic Behavior

• Trigonometric functions exhibit periodic behavior, meaning they repeat values at regular intervals
• The period is the smallest positive value p for which $f(x + p) = f(x)$ for all values of x
• Sine and cosine functions have a period of $2π$ radians or 360°
  • After rotating 360° around the unit circle, sin θ and cos θ will repeat the same values
• The tangent and cotangent functions have a period of $π$ radians or 180°
  • Tan θ and cot θ will repeat values after rotating 180° around the unit circle

Calculating Trigonometric Function Values

• Secant and cosecant functions, like their reciprocals cosine and sine, also have a period of $2π$ radians or 360°
• Periodicity allows for the calculation of trigonometric function values for angles greater than 360° or less than 0°
  • This is done by finding the coterminal angle within the first rotation of the unit circle

Trigonometric Functions for Special Angles

Exact Values without a Calculator

• Special angles are angles for which the trigonometric function values can be exactly determined without a calculator
  • These angles include 0°, 30°, 45°, 60°, 90°, and their multiples
• At 0°, $(x,y) = (1,0)$, so cos 0° = 1, and sin 0° = 0
  • Tan 0° = 0, cot 0° is undefined, sec 0° = 1, and csc 0° is undefined
• At 30°, $(x,y) = (\sqrt{3}/2, 1/2)$, so cos 30° = $\sqrt{3}/2$, sin 30° = 1/2, and tan 30° = $1/\sqrt{3}$
  • Cot 30° = $\sqrt{3}$, sec 30° = $2/\sqrt{3}$, and csc 30° = 2
• At 45°, $(x,y) = (1/\sqrt{2}, 1/\sqrt{2})$, so cos 45° = $1/\sqrt{2}$, sin 45° = $1/\sqrt{2}$, and tan 45° = 1
  • Cot 45° = 1, sec 45° = $\sqrt{2}$, and csc 45° = $\sqrt{2}$

Reference Angles and Quadrant Rules

• At 60°, $(x,y) = (1/2, \sqrt{3}/2)$, so cos 60° = 1/2, sin 60° = $\sqrt{3}/2$, and tan 60° = $\sqrt{3}$
  • Cot 60° = $1/\sqrt{3}$, sec 60° = 2, and csc 60° = $2/\sqrt{3}$
• At 90°, $(x,y) = (0,1)$, so cos 90° = 0, sin 90° = 1, and tan 90° is undefined
  • Cot 90° = 0, sec 90° is undefined, and csc 90° = 1
• Trigonometric function values for angles in the other quadrants can be determined using reference angles and the unit circle quadrant sign rules