Terms

• Function Hexagon consists of six trigonometric ratios namely: sine ratio, cosine ratio, tangent ratio, cotangent ratio, secant ratio, and cosecant ratio
• The functions under the function hexagon are Reciprocal, Quotient, Product, Cofunction, and Pythagorean Identities
• Reciprocal Identities - where: sin 𝜃 = 1/csc𝜃 , csc 𝜃 = 1/sin 𝜃 , cos 𝜃 = 1/sec𝜃, sec 𝜃 = 1/cos 𝜃 , tan 𝜃 = 1/cot 𝜃 , and cot 𝜃 = 1/tan𝜃
• Quotient Identities - for clockwise rotation, tan 𝜃 = sin𝜃/cos 𝜃 , sin 𝜃 = cos 𝜃/cot 𝜃 , cos 𝜃 = cot 𝜃/csc𝜃 , cot 𝜃 = csc𝜃/sec𝜃 , csc 𝜃 = sec 𝜃/tan𝜃 , and sec 𝜃 = tan𝜃/sin 𝜃 .
• Quotient Identities - for counterclockwise rotation, tan 𝜃 = sec𝜃/csc𝜃 , sec 𝜃 = csc𝜃/cot 𝜃 , csc 𝜃 = cot𝜃/cos 𝜃 , cot 𝜃 = cos 𝜃/sin 𝜃 , cos 𝜃 = sin 𝜃/tan𝜃 , and sin 𝜃 = tan𝜃/sec 𝜃
• Product Identities - where: sin 𝜃 ∙ csc 𝜃 = 1, cos 𝜃 ∙ sec 𝜃 = 1, and tan 𝜃 ∙ cot 𝜃 = 1
• Cofunction Identities - where sin 𝜃 = cos(90° − 𝜃), tan 𝜃 = cot(90° − 𝜃), and sec 𝜃 = csc(90° − 𝜃)
• Pythagorean Identities - where 𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠^2𝜃 = 1, 𝑡𝑎𝑛^2𝜃 + 1 = 𝑠𝑒𝑐^2𝜃, and 1 + 𝑐𝑜𝑡^2𝜃 = 𝑐𝑠𝑐^2𝜃
• A unit circle is a circle whose center is at (0,0) and whose radius is 1 unit. It is represented by the equation x^2 + y^2 = 1. Since the circumference of a circle is 2pi, then the 2πr circumference of a unit circle is 2π units or approximately 6.28 units. A unit circle in a Cartesian plane is divided into four congruent arcs.
• The domain (values of θ) of sine and cosine functions is the set of all real numbers.
• Domain: − 1≤y≤1 and − 1≤x≤1. Therefore, the values of sin sin θ and

  cos cos θ also ranges between − 1 and 1 inclusive.
• One complete revolution is 360° or 2π.
• Functions that act in such a cyclic or repetitive manner are called periodic functions.
• A sine/cosine function is periodic and its graph is known as the sine/cosine curve.
• Sine/Cosine Function:

  
  

  1. Its domain is the set of all numbers because the graph extends indefinitely on both sides.
  2. The range is {−1 ≤ y ≤ 1} for the graph never rises above 1 or falls below -1.
  3. The period is 2π because the pattern repeats in every 2π interval.
  4. The sine function is odd and its graph is symmetric with respect to the origin.
  5. The cosine function is even and its graph is symmetric with respect to y-axis.
• Period is the length of the smallest interval that contains exactly one copy of the repeating pattern.
• Period formula: $2pi/lBl$
• The amplitude is the distance from the midline to the maximum point or to the lowest point of the sine or cosine graph.
• The amplitude of y = A sinBx and y = A cosBx is 1 2 (M − m) where M is the maximum point and m is the minimum point both given by the range.
• When sine or cosine function is moved to the left or right along the x −axis, the graph is horizontally shifted or horizontally translated. In other words, this shift or translation is referred to as phase shift.
• Phase shift in y = A sinB(x − h) or y = A sinB(x − h) is h where a positive h indicates that the usual graph is shifted to the right.
• When sine or cosine function is moved upwards or downwards along y −axis, the graph is vertically shifted or vertically translated. In this case, the midline, y = k is shifted or translated. The vertical shift in y = A sinB(x − h) + k or y = A cosB(x − h) + k is k where a positive k indicates that the usual graph is shifted upwards. The midline of the graph is also given by the average of the minimum and the maximum points (y −coordinates) that is 1 2 (M + m).
• The slope of the terminal side of an angle, θ, in radians in standard position is tan x. It is undefined when angle, x = ± π 2 thus the terminal side of such angle is vertical. Such vertical line is called an asymptote. It is the line where the graph of tan x approaches but never touches.
• The tangent function is periodic with period π: For every real number x in its domain tan(x ± π) = tan x
• Cosecant function, at a given value of x, is undefined when sine of x is equal to 0. Thus, vertical asymptotes of the graph of cosecant function where sin x = 0 is at x = nπ, where n is an integer and sine is zero at these x −values. Similarly, the secant function, with cos x = 0 does not exist. In other words, the graph of secant function has vertical asymptotes at x = π 2 + nπ.
• csc x = 1/sin x and sec x = 1/cos x
• A = amplitude

  B = determines the period

  Period(P) = tangent and cotangent ( π/B ); secant and cosecant ( 2π/B)

  h = phase shift

  k = vertical shift