Trigonometry - Trigonometric Functions - Quick Guides

Page 1 of 36

Trigonometry - Trigonometric Functions - Page 1 preview image

Study GuideTrigonometry–Trigonometric Functions1. Angles1.1What Is an Angle?Anangleis a measure of rotation. It tells us how much one line has turned away from another.Angles are measured indegrees(°).•One full rotation around a point equals360°.An angle is formed by two rays:•Theinitial sideis where the rotation starts.•Theterminal sideis where the rotation ends.1.2Positive and Negative AnglesThe direction of rotation matters.•Positive anglesare formed by rotatingcounterclockwise.•Negative anglesare formed by rotatingclockwise.So, whether an angle is positive or negative depends only on thedirection of rotation, not its size.1.3Standard PositionFigure 1(a) A positive angle and(b) a negative angle.

Page 2 of 36

Trigonometry - Trigonometric Functions - Page 2 preview image

Study GuideAn angle is said to be instandard positionwhen:•Itsinitial side lies along the positive x-axis.•The vertex of the angle is at theorigin(0, 0).Placing angles in standard position helps us describe and compare them more easily.1.4Quadrantal AnglesFigure 2 Types of Angles

Page 3 of 36

Trigonometry - Trigonometric Functions - Page 3 preview image

Study GuideSome angles have theirterminal side exactly on one of the coordinate axes.These are calledquadrantal angles.Examples include:•0°•90°•180°•270°•Any angle whose terminal side lies directly on the x-axis or y-axis1.5Angles in the Four QuadrantsIf an angle in standard position isnot quadrantal, then its terminal side lies inone of the fourquadrantsof the coordinate plane.Here’s how angles are classified based on where the terminal side ends up:•First Quadrant: Terminal side is between the positive x-axis and positive y-axis•(Example: 40°)•Second Quadrant: Terminal side is between the positive y-axis and negative x-axis•(Example: −260°)•Third Quadrant: Terminal side is between the negative x-axis and negative y-axis•(Example: 210°)•Fourth Quadrant: Terminal side is between the negative y-axis and positive x-axis•(Example: −50°)

Page 4 of 36

Trigonometry - Trigonometric Functions - Page 4 preview image

Study Guide1.6Understanding Angles in Standard PositionWhen an angle is drawn instandard position, itsvertex is at the origin(0, 0)and itsinitial sidelies along the positive x-axis. From there, the angle opens eithercounterclockwise(positiveangles)orclockwise(negative angles).Where theterminal sideof the angle ends tells us whichquadrantthe angle is in.Example 1:Identifying the Quadrant of an Angle

Page 5 of 36

Trigonometry - Trigonometric Functions - Page 5 preview image

Study GuideFigure 3 Angles coterminal with 30°1.7What Does Coterminal Mean?Two angles are calledcoterminalif:•They are instandard position, and•They share thesame terminal side.In other words, they may look different, but theyend in exactly the same place.

Page 6 of 36

Trigonometry - Trigonometric Functions - Page 6 preview image

Study Guide1.8Coterminal Angles and Full RotationsA full rotation around a circle is360°.So, you can create coterminal angles byadding or subtracting 360°.General Rule:All angles that are coterminal with an angle(d°)can be written as:where(n)is any integer(positive, negative, or zero).Example: Angles Coterminal with 30°All the angles below share the same terminal side as30°, even though they look very different:Even though these angles involve multiple rotations, they all end in thesame position.1.9What Does “Coterminal” Mean?Two angles are calledcoterminal anglesif they:•Start at the same initial side•End at the same terminal sideThis happens when the angles differ by awhole number multiple of 360°.In general:wherenis any integer(positive, negative, or zero).Example 2:Are 200° and 940° Coterminal?To check if two angles are coterminal, subtract one from the other and see if the result is a multiple of360°.If 940° and 200° were coterminal:

Page 7 of 36

Trigonometry - Trigonometric Functions - Page 7 preview image

Study GuideSubtract 200° from both sides:Since740 is not a multiple of 360, there is no whole number value ofn.Conclusion:200° and 940° are not coterminal angles.Example 3:Name Five Angles Coterminal with −70°To find coterminal angles, add or subtract multiples of 360°.Starting with −70°:•−70° +(1 × 360°) =290°•−70° +(2 × 360°) =650°•−70° +(3 × 360°) =1010°•−70° +(−1 × 360°) =−430°•−70° +(−2 × 360°) =−790°All of these angles share the same terminal side as −70°, so they are coterminal.1.10Degrees, Minutes, and SecondsAngle measurements are not always whole degrees. Fractional angles can be written in two mainways:•Asdecimal degrees, such as34.25°•Usingdegrees, minutes, and seconds(DMS)1.11Understanding Minutes and SecondsThe degree is divided into smaller units:•1 degree(°) = 60 minutes(′)•1 minute(′) = 60 seconds(″)

Page 8 of 36

Trigonometry - Trigonometric Functions - Page 8 preview image

Study GuideIn symbols:•1° = 60′•1′ = 60″1.12Converting Between FormsExample 4:Write 34°15′ Using Decimal DegreesConvert minutes to degrees by dividing by 60.Example 5:Write 12°18′44″ Using Decimal DegreesConvert minutes and seconds step by step.Example 6:Write 81.293° Using Degrees, Minutes, and SecondsStart with the whole number part:Convert the decimal part to minutes:So:Convert the decimal minutes to seconds:

Page 9 of 36

Trigonometry - Trigonometric Functions - Page 9 preview image

Study GuideFinal Answer:Summary•Coterminal angles differ by multiples of360°•Minutes and seconds are smaller units of degrees•You can convert between decimal degrees and DMS by multiplying or dividing by60•These conversions are essential for accuracy in geometry and trigonometry2. Functions of Acute Angles2.1Why Similar Triangles MatterTrigonometry is built on the idea ofsimilar triangles, first studied by Euclid.Two triangles aresimilarif:•Their corresponding angles are equal, and•Their corresponding sides are in the same ratio.Becauseall right triangles have a 90° angle, any two right triangles that share another equal anglemust be similar. This means theratios of their corresponding sides are the same, even if thetriangles are different sizes.This idea leads directly totrigonometric ratios.2.2Naming AnglesAngles are usually represented usinglowercase Greek letters.The most common ones are:•α(alpha)•θ(theta)The letter used doesn’t change the math—it’s just a name for the angle.

Page 10 of 36

Trigonometry - Trigonometric Functions - Page 10 preview image

Study Guide2.3Measuring AnglesAngles can be measured indegreesorradians.The relationship between them is:•180° = πradians•1° = π / 180radians•1 radian = 180° /πYou’ll often switch between these units in trigonometry, so it’s important to know these conversions.2.4Triangles and the CircleTo define trigonometric ratios, we often use aright triangleplaced inside a circle with equation:Figure 1 Reference triangles.In this triangle:•ris the hypotenuse•xis the side adjacent to the angle•yis the side opposite the angle

Preview Mode

This document has 36 pages. Sign in to access the full document!

Report

Learning Tools

Writing Tools

Browse Resources

Trigonometry - Trigonometric Functions - Page 1

Trigonometry - Trigonometric Functions

Study Now!

Document Details

Related Documents

Trigonometry - Vectors

Trigonometry - Trigonometry of Triangles

Trigonometry - Trigonometric Identities

Trigonometry - Polar Coordinates and Complex Numbers

Trigonometry - Inverse Functions and Equations

Trigonometry - Graphs of Trigonometric Functions

Trigonometry - Additional Topic

Precalculus - Polynomial and Rational Functions

Precalculus - Functions

Precalculus - Exponential and Logarithmic Functions

Company

Explore

Study Tools