DAT - Quantitative Reasoning, Trigonometry

SOH-CAH-TOA

Sine = Opposite/Hypotenuse; Cosine = Adjacent/Hypotenuse; Tangent = Opposite/Adjacent

tangent

the length of the opposite side over the length of the adjacent side - the tangent of the angle θ is written as tan θ

cosine

the length of the side adjacent to angle θ (actually, there will be two sides adjacent to the angle, but one of those sides will be the hypotenuse, so by adjacent, we really mean the side adjacent to angle that is not the hypotenuse) divided by the length of the hypotenuse - the cosine of the angle θ is written as cos θ

sine of angle θ

the length of the side opposite the angle divided by the length of the hypotenuse - the sine of the angle θ is written sin θ

sin² x + cos² x

= 1

1 + cot² x

= csc² x

tan² x + 1

= sec² x

periodic function

The trigonometric function f(x) = sin x repeats itself every 360° (e.g., sin 30° = sin 390°) - let R be a positive constant. If f is a function such that f(x) = f(x + R) for all x, and R is the smallest positive number for which this is true, then the function is said to be periodic with period R. Notice that the function f defined by f(x) = sin x, sin (x + 2π) = sin x for all x. Because 2π is the smallest possible positive value for R for which sin (x + R) = sin x for all x, 2π is the period of sin x. While it is also true that sin (x + 4π) = sin x for all x, 4π is not the smallest possible positive value for R such that sin (x + R) = sin x for all x - again, the period of sin x is 2π

function (f)

a set of instructions that associates each number of a set A (which is called the domain) with a number in a set B (which is called the range). Often the domain and/or range are not specified. What is important is that you follow the instructions: For each given number, use the function to find the number associated with it - if x is a point of the domain, then f(x) is the number associated with x by the function f. Notice the difference between f and f(x) while f refers to the entire process of using the instructions to associate numbers with other numbers, f(x) means the exact number that the function f associates with the number x. So f(x) is a number for any specified x

radian

another measure used to describe an angle and is often used in trigonometry - in a circle, a central angle is the angle formed by two radii of the circle. This leads to the definition of a radian. If a central angle of a circle intercepts an arc of a circle with length l, then the number of radians in the central angle that contain this arc is l/r

trigonometry

focuses on right triangles

Pythagorean theorem

in a right triangle, a² + b² = c². Now divide both sides of this equation by c² to get (a²/c²) + (b²/c²) = 1 --> (a/c)² + (b/c)² = 1. Now a/c = sin θ and b/c = cos θ. So sin² θ + cos² θ = 1

graph of y = cos x

same graph as y = sin x, except at x = 0, sin x = 1

tan 45°

tan 45° = 1 because this is where sine equals cosine. A tangent larger than on occurs when sine is larger (the angle is larger than 45° and smaller than 90°), and a tangent smaller than 1 occurs when cosine is larger (the angle is smaller than 45° and larger than 0°)

graph of y = tan x

tan x is undefined for every odd multiple of 90° or π/2

converting radians and degrees

the circumference C of a circle is related to its radius by C = 2πr. The number of radians in the circumference any any circle is 2π because there are 2π radii in the length of the circumference of any circle. However, there are also 360° in any full circle, so 2π radians is equal to 360°. Therefore, 1 radian is equal to 360/2π = 180/π degrees

graph of y = sin x

the graph of y = sin x repeats itself every interval of length 360 degrees (or 2π radians) - when 0 ≤ x ≤ 90°, sin x increases from 0 (x = 0°) to 1 (x = 90°). When 90 ≤ x ≤ 180°, sin x decreases from 1 to 0 (x = 180°). When 180 ≤ x ≤ 270°, sin x decreases from 0 to -1 (x = 270°). When 270 ≤ x ≤ 360°, sin x increases from -1 to 0 (x = 360°)

inverse functions

the inverse sine function of x, which is denoted by arcsin x, or sin⁻¹ x, determines the angle y such that sin y = x and -90° ≤ y ≤ 90°. Because -1 ≤ sin y ≤ 1, the inverse sine function cannot be defined for value of x such that x < -1 or x > 1. Therefore, the inverse sine function is defined only for x such that -1 ≤ x ≤ 1 (e.g., sin⁻¹ (1/2) = 30° because sin 30° = 1/2 and 30° is in the interval -90° ≤ y ≤ 90°

secant (sec θ)

the reciprocal of cosine and is equal to Hypotenuse/Adjacent = 1/cos θ

cosecant (csc θ)

the reciprocal of sine and is equal to Hypotenuse/Opposite = 1/sin θ

cotangent (cot θ)

the reciprocal of tangent and is equal to Adjacent/Opposite = 1/tan θ