Precalculus Chapter 6: Trigonometric Functions

Linear speed

v = s/t s is distance traveled (arc length) in time t around the circle

Angular speed

w = θ/t, the w is the Greek letter omega

Transformation Graph of y = Asin(wx)

|A| = Amplitude; Vertical Stretch/Compression Period = (2pi)/w; Horizontal Stretch/Compression

45-45-90 Right Triangle

-This right triangle is an isosceles right triangle -If each of the sides that form the right angle has a measure of 1, then using the Pythagorean theorem, you find that the hypotenuse has the value √2 -If an isosceles right triangle had each of the equal sides with a measure of 5, then the hypotenuse would have a measure of 5√2

Steps for Graphing Sine & Cosine Transformation Graphs

1. Determine the amplitude and period of the function. 2. Divide the period into 4 sub-intervals of the same length. 3. Make a table of points. Use the endpoints of each sub-interval to obtain five key points on the graph. 4. Plot the point on the periodic coordinate planes and draw the function for at least once cycle.

Unit Circle

A circle with a radius of 1, centered at the origin of a coordinate plane, where point P along that circle contains the coordinates (x, y). Through use of the pythagorean theorem and the definitions of sine and cosine, common ordered pairs are found.

right triangle

A triangle that has a 90 degree angle.

Domain of the trigonometric functions

Domain of sine and cosine are all real numbers. Domain of tangent and secant are all real numbers except odd integer multiples of π/2 (90°). Domain of cotangent and cosecant are all real numbers except integer multiples of π(180°)

Fundamental Identities

Given P (x, y) is the point on the unit circle corresponding to angle theta. Furthermore, sin (theta) = y and cos (theta) = x

Quotient Identities

Given P (x, y) is the point on the unit circle corresponding to angle theta. Furthermore, sin (theta) = y and cos (theta) = x

Reciprocal Identities

Given P (x, y) is the point on the unit circle corresponding to angle theta. Furthermore, sin (theta) = y and cos (theta) = x

standard position of an angle

In a coordinate plane, the position of an angle whose vertex is at the origin and whose initial side lies on the positive x-axis. The initial side remains fixed and the terminal side can rotate an infinite amount of times.

Cosecant

In a right triangle it is the ratio of the length of the hypotenuse to the leg opposite from an angle.

Sine

In a right triangle it is the ratio of the length of the leg from an angle to the hypotenuse.

Secant (triangle)

In a right triangle, the ratio of the hypotenuse to the length of the leg adjacent from the angle

Cosine

In a right triangle, the ratio of the length of the leg adjacent from the angle to the length of the hypotenuse

Cotangent

In a right triangle, the ratio of the length of the leg adjacent from the angle to the length of the leg opposite from the angle.

Tangent (triangle)

In a right triangle, the ratio of the length of the leg opposite from the angle to the length of the leg adjacent from the angle.

Convert from radians to degrees

Multiply by 180 and divide by pi

Convert from degrees to radians

Multiply by pi and divide by 180

Range of the trigonometric functions

Range of sine and cosine is [-1,1] Range of tangent and cotangent are all real numbers Range of cosecant and secant are all real numbers that are less than negative 1 or greater than positive 1.

secant graph

Reciprocal of the cosine graph.

cosecant graph

Reciprocal of the sine graph.

cotangent graph

Reciprocal of the tangent graph.

angle of depression

The angle formed by a horizontal line and the line of sight to an object below the horizontal line

Area of a sector of a circle

The area of part of a circle is found by multiplying the area of the whole circle by the fraction of the circle that is of interest. The fraction of the circle is dependent upon the central angle.

tangent graph

This is the parent function of tangent with a period of pi and there is no amplitude. Note the parent function's period occurs in the domain (-pi/2, pi/2) and there are vertical asymptotes.

Amplitude

The height of a periodic function. Typically notated by |A|.

Radian

The measure of a central angle that intercepts an arc with length equal to the radius of the circle

30-60-90 Right Triangle

The sides are in a ratio of shortest side, x, opposite the 30 degree angle medium side, x√3, opposite the 60 degree angle longest side, 2x, opposite the 90 degree angle

Pythagorean Theorem

The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.

Graph of y = cos x

This is the parent function of cosine with a period of 2pi and an Amplitude of 1. Note the parent function's period occurs in the domain [0, 2pi].

Graph of y = sin x

This is the parent function of sine with a period of 2pi and an Amplitude of 1. Note the x-axis of our coordinate plane is now an angle measures in radians and the y-axis is the sine of that angle measure. Also notice the image shows 2 periods of the function. The parent function's period occurs in the domain [0, 2pi].

Period of the trigonometric functions

This refers to the radian measure that is needed before the values of a given function repeat (or cycle). Sine, cosine, secant, cosecant have a period of 2π (θ + 2πk). Tangent and cotangent have a period of π (θ + πk), where k is any integer.

Periodic Properties

Use them to find exact values. For example, sin(17π/4) is also sin(π/4 + 4π). So, by the property sin(π/4) = √2/2

Arc Length of a Circle

s = rθ, where s = arc length, r = radius and theta = the angle measure in radians

angle of elevation

the angle formed by a horizontal line and the line of sight to an object above the horizontal line