Trigonometric Functions
Modeling with Periodic Functions
How Can Sine And Cosine Help You Solve Real-World Problems Involving Cycles? How To Model With A Sine Function A tuning fork that plays the A above middle C vibrates with a frequency of 440 Hz. If the tuning fork has a maximum displacement of 0.5 mm, derive a sine model for the tuning fork that gives the displacement d(mm) as a function of time t(sec). (t,d) General equation: d=asin(bt)d=\asin(bt) Amplitude: a = 0.5 mm Period: ƒreq= 1 period = b 2π freq=\frac{1}{period}=\frac{b}{2\pi} b=880πb=880\pi Model: d=0.5sin(880π)d=0.5\sin(880\pi) How to Model with a Cosine Function A weight attached to a spring is at its lowest point 6 inches below the equilibrium point at time t = 0 seconds. When the weight is released, it oscillates and returns to its original position at t = 5 seconds. Write an equation to model the distance, d, from equilibrium after t seconds. General equation: d=acos(bt)d=\acos(bt) Amplitude: a=∣−b∣a=\mid-b\mid Period: 5= 2π b → b= 2π 5 5=\frac{2\pi}{b}\:\longrightarrow\:b=\frac{2\pi}{5} Model: d=−6cos( 2π 5 t)d=-6\cos(\frac{2\pi}{5}t)
Angles in Standard Position
How Can We Define and Analyze Angles Using Their Characteristics? Coterminal side: the angles in standard position with the same terminal side Standard position: the position of an angle with its vertex at the origin and the initial side on the positive x-axis Terminal side: the side of an angle, or ray, that is rotated about the vertex Degree Measure Degrees are used to measure the size of an angle. One complete revolution 1°= 1 360 revolution Anatomy of an Angle The terminal side of an angle is the side of an angle, or the ray, that is rotated about the vortex. (Counter-clockwise) The negative side of an angle is on the clockwise side of the initial side. Angles in the Coordinate Plane You can draw angles in any coordinate plane as long as they have two rays, and a vertex. (Any quadrant) An angle is said to be in standard position if it has its vertex at the origin and the initial side on the positive x-axis. The other ray is called the terminal side Co-terminal Angles Two angles in standard position are said to be co-terminal if they have the same terminal side. Represent All Co-terminal Angles List the measures of the angles co-terminal with a 90 degree angle. To do this, you can simply add and subtract 360 degrees from 90, and continually get co-terminal angles. You can never find all of them of course as they are an infinite amount of numbers. Angles with measure (m+360n)°(m+360n)^\circ are co-terminal with a m°m^\circ angle, where m is any real number and n is any integer. For this problem, (90+360n) will give you all the co-terminal angles.
Right Triangle Trigonometry
How Can You Find Unknown Side Lengths And Angle Measures Of Right Triangles? Relation of Sides to a Given Angle The hypotenuse is always the longest side, opposite the right angle. There are also the adjacent and opposite angle. Sin(A) = opposte leg hypotenuse \frac{opposte\:leg}{hypotenuse} Cos(A) = adjacent leg hypotenuse \frac{adjacent\:leg}{hypotenuse} Tan(A) = opposite leg adjacent leg \frac{opposite\:leg}{adjacent\:legg\:} SOH CAH TOA How to Use Trigonometric Ratios to Find an Unknown Side Length Identify an angle to use. Name the sides based on that angle. Write an equation using a trigonometric ratio. Solve the equation. You Must Have Your Calculator In Degrees Mode How to Use a Trigonometric Ratio to Find an Unknown Angle Measure Find the value of y to the nearest degree. tan(y)= 18 9.3 \tan(y)=\frac{18}{9.3} →\longrightarrow y=tan−1( 18 9.3 )y=\tan^{-1}(\frac{18}{9.3}) = 62.68 degrees. Special Right Triangles For any 45-45-90 triangle, if the legs have length x, then the hypotenuse has length x√2x\sqrt{2} For any 30-60-90 triangle, the legs have lengths x and x√3x\sqrt{3} , and the hypotenuse has length 2x. Ex: (Notes) How to Solve a Right Triangle Solve the triangle (notes) Use the Pythagorean Theorem to find x. Use trigonometry to find one of the unknown angle measures. Use the sum of the angles to find the other unknown angle measure.
Evaluating the Six Trigonometric Functions
How Can You Use The Relationships Among Trigonometric Functions To Evaluate Them? Evaluate Trigonometric Functions Given a Point on the Terminal Side For unit circle: cos(θ)=x... sin(θ)=y\cos(\theta)=x...\:\sin(\theta)=y For other angles not on the unit circle: cos(t)= x r ... sin(t)= y r \cos(t)=\frac{x}{r}\:...\:\sin(t)=\frac{y}{r} Evaluate Trigonometric Functions Given a Point on the Terminal Ray (Notes) Developing the Pythagorean Identities Pythagorean identities: 1.sin2(θ)+cos2(θ)=1\sin^2(\theta)+\cos^2(\theta)=1 tanθ= sinθ cosθ \tan\theta=\frac{\sin\theta}{\cos\theta} cotθ= cosθ sinθ \cot\theta=\frac{\cos\theta}{\sin\theta} (Not a Pythagorean Identity) 2. tan2(θ)+1=sec2(θ)\tan^2(\theta)+1=\sec^2(\theta) 3. 1+cot2(θ)=csc2(θ)1+\cot^2(\theta)=\csc^2(\theta) You can square all values just like the first equation How To Use Identities To Find Trigonometric Values Choose an appropriate Pythagorean identity. Substitute known value. Solve. Determine the sign. Given that sin(θ)= 4 5 and π 2 <θ<π\sin(\theta)=\frac{4}{5}\:and\:\frac{\pi}{2}<\theta<\pi find the remaining trigonometric values. 1. sin2(θ)+cos2(θ)=1\sin^2(\theta)+\cos^2(\theta)=1 2. 16 25 +cos2(θ)=1\frac{16}{25}+\cos^2(\theta)=1 3. cos2(θ)= 9 25 → √cos2(θ)=√ 9 25 \cos^2(\theta)=\frac{9}{25}\:\longrightarrow\:\sqrt{\cos^2(\theta)}=\sqrt{\frac{9}{25}} 4. cos(θ)=± 3 5 → − 3 5 (Quadrant II)\cos(\theta)=\pm\frac{3}{5}\:\longrightarrow\:-\frac{3}{5}\:(Quadrant\:II) Other trigonometric values: csc(θ)= 5 4 \csc(\theta)=\frac{5}{4} sec(θ)=− 5 3 \sec(\theta)=-\frac{5}{3} tan(θ)= sinθ cos(θ) =( 4 5 − 3 5 )=( 4 5 )(− 5 3 )=− 20 15 =− 4 3 \tan(\theta)=\frac{\sin\theta}{\cos(\theta)}=(\frac{\frac{4}{5}}{-\frac{3}{5}})=(\frac{4}{5})(-\frac{5}{3})=-\frac{20}{15}=-\frac{4}{3} cot(θ)=− 3 4 \cot(\theta)=-\frac{3}{4}
Changes in Period and Phase Shift of Sine and Cosine Function
How Is A Transformation Of The Graph Related To The Equation Of A Sine Or Cosine Function? Frequency: the number of cycles of a periodic function that occur in one horizontal unit Phase shift: the horizontal translation of a periodic function Period and Frequency Period: The horizontal length of one cycle of a periodic function Frequency: The number of cycles of a periodic function that occur in one horizontal unit If the value of b is more than one, the period becomes greater. If it is less than one, the period gets smaller. Period, Frequency, and Frequency Factor Period Formula: 2π b \frac{2\pi}{b} (Radians) 360 b \frac{360}{b} (Degrees) Frequency Formula: b 2π \frac{b}{2\pi} (Radians) b 360 \frac{b}{360} (Degrees) Horizontal Translations as Phase Shifts Phase shift: A horizontal translation of a periodic function y=cos(x) → y=cos(x− π 2 )y=\cos(x)\:\longrightarrow\:y=\cos(x-(\frac{\pi}{2})) By adding a value to a graph that is the same as its period, the output will coincide with the original graph. Therefore: 0≤h≤ 2π b 0\le h\le\frac{2\pi}{b} Transformations of Sine and Cosine y=asinb(x−h)+ky=\asin b(x-h)+k ∣a∣=amplitude\mida\mid=amplitude 2π b ... 360° b =period\frac{2\pi}{b}\:...\:(\frac{360^\circ}{b})=period h = horizontal shift, or phase shift k = vertical shift
The Unit Circle
The Unit Circle Can Trigonometric Functions Be Extended To Any Angle Measure? Quadrantal angle: an angle that terminates on the x- or y-axis when in standard position (angles that measure 0°, 90°, 180°, 270°, and their co-terminal angles) Reference angle: the acute angle between the terminal side of an angle in standard position and the x-axis Unit circle: a circle with radius 1, centered at the origin Radians and Degrees π rad=180\pi\:rad=180 180/6 = 30 210°=7(30°) → =7( π 6 ) = 7π 6 210^\circ=7(30^\circ)\:\longrightarrow\:=7(\frac{\pi}{6})\:=\frac{\:7\pi}{6} 5π 3 =π+ 2 3 π → 180°+120°=300°\frac{5\pi}{3}=\pi+\frac{2}{3}\pi\:\longrightarrow\:180^\circ+120^\circ=300^\circ Important degree measures: π 2 =90°\frac{\pi}{2}=90^\circ π 3 =60° \frac{\pi}{3}=\frac{60^\circ,\:\pi}{ }\: π 4 =45°\frac{\pi}{4}=45^\circ π 6 =30°\frac{\pi}{6}=30^\circ The Unit Circle A unit circle is a circle with radius 1, centered at the origin. Define the Trigonometric Functions on the Unit Circle If the terminal side of an angle in standard position intersects the unit circle at (x,y), then: cosθ=x\cos\theta=x and sinθ=y\sin\theta=y and tanθ= y x \tan\theta=\frac{y}{x} How to Find Coordinates of a Point on the Unit Circle What are the coordinates of the point where the terminal side of an angle measuring π 3 \frac{\pi}{3} radians intersects the unit circle? (x,y) = (cos π 3 , sin π 3 )(\cos(\frac{\pi}{3}),\:\sin(\frac{\pi}{3})) = ( 1 2 , √3 2 )(\frac{1}{2},\:(\frac{\sqrt{3}}{2})) The Quadrantal Angle A quadrantal angle is an angle in standard position that terminates on the x- or y-axis (1,0): 0\circ,360\circ, radians: 0, 2π0^{\circ},360^{\circ},\:radians:\:0,\:2\pi (0,1): 90°, π 2 radians90^\circ,\:\frac{\pi}{2}\:radians cos( π 2 )=0\cos(\frac{\pi}{2})=0 sin( π 2 )=1\sin(\frac{\pi}{2})=1 tan( π 2 )= 0 1 =undeƒined\tan(\frac{\pi}{2})=\frac{0}{1}=un-defined (-1,0): 180°, π radians180^\circ,\:\pi\:radians cosπ=−1 ... sinπ=0 ... tanπ=0\cos\pi=-1\:...\:\sin\pi=0\:...\:\tan\pi=0 (0,-1): 270°, 3π 2 radians270^\circ,\:\frac{3\pi}{2}\:radians cos( 3π 2 )=0 ... sin( 3π 2 )=−1 ... tan( 3π 2 )=undeƒined\cos(\frac{3\pi}{2})=0\:...\:\sin(\frac{3\pi}{2})=-1\:...\:\tan(\frac{3\pi}{2})=undefined θ:\theta: domain: −∞→∞-\infty\longrightarrow\infty range: −1→1-1\longrightarrow1 Reference Angle The reference angle is the acute angle formed by the terminal side of an angle in standard position and the x-axis. It cannot be negative. Quadrant Containing Terminal Side of θ:\theta: Reference Angle, ϕ\phi I: θ\theta II: 180°−θ, or π−θ180^\circ-\theta,\:or\:\pi-\theta (degrees, radians) III: θ−180°, or θ−π\theta-180^\circ,\:or\:\theta-\pi (degrees, radians) IV: 360°−θ, or 2π−θ360^\circ-\theta,\:or\:2\pi-\theta (degrees, radians) How to Identify a Reference Angle What is the reference angle for θ= 5π 4 ?\theta=\frac{5\pi}{4}? ϕ=θ−π\phi=\theta-\pi 5π 4 − 4π 4 = π 4 \frac{5\pi}{4}-\frac{4\pi}{4}=\frac{\pi}{4} That is our reference angle in terms of radians, or 45°45^\circ in terms of degrees. Determine Signs of Trigonometric Function Values All Students Take Calculus I: sinθ>0...cosθ>0...tanθ>0\:\sin\theta>0...\cos\theta>0...\tan\theta>0 All II:sinθ>0\sin\theta>0 Only Sin III: tanθ>0\tan\theta>0 Only Tangent IV: cosθ>0\cos\theta>0 Only Cos How to Evaluate Trigonometric Functions in Any Quadrant Identify the reference angle, ϕ\phi Find the value of the function for ϕ\phi Determine the sign of the value based on the quadrant containing the terminal side What is the exact value of cos(150°)?\cos(150^\circ)? ϕ=30°\phi=30^\circ cos(30\circ)= √3 2 \cos(30^{\circ})=\frac{\sqrt{3}}{2} cos(150°)=− √3 2 \cos(150^\circ)=-\frac{\sqrt{3}}{2}
Reciprocal Trigonometric Functions
What Are The Reciprocal Trigonometric Functions? Reciprocal Trigonometric Functions SOH CAH TOA csc(A)= 1 sin(A) = hypotenuse opposite leg \csc(A)=\frac{1}{\sin(A)}=\frac{hypotenuse}{opposite\:leg} sec(A)= 1 cos(A) = hypotenuse adjacent leg \sec(A)=\frac{1}{\cos(A)}=\frac{hypotenuse}{adjacent\:leg} cot(A)= 1 tan(A) = adjacent leg opposite leg \cot(A)=\frac{1}{\tan(A)}=\frac{adjacent\:leg}{opposite\:leg} Trigonometric Values from a Given Value An angle, θ\theta , of a right triangle has a cosecant value of 2. Find the remaining trigonometric values for θ.\theta. (notes) How to Simplify Trigonometric Expressions Simplify: sec(θ)cos(θ)\sec(\theta)\cos(\theta) = 1 cos(θ) •cos(θ)=1=\frac{1}{\cos(\theta)}\bullet\cos(\theta)=1 Simplify: cot(θ) cos(θ) \frac{\cot(\theta)}{\cos(\theta)} = cos(θ) sin(θ) cos(θ) =\frac{\cos(\theta)}{\frac{\sin(\theta)}{\cos(\theta)}} = cos(θ) sinθ • 1 cos(θ) =\frac{\cos(\theta)}{\sin\theta}\bullet(\frac{1}{\cos(\theta)}) = 1 sinθ =\frac{1}{\sin\theta} =csc(θ)=\csc(\theta) Reciprocal Trigonometric Functions and the Unit Circle csc(θ)= 1 y , y≠0\csc(\theta)=\frac{1}{y},\:y\ne0 sec(θ)= 1 x ,x≠0\sec(\theta)=\frac{1}{x},x\ne0 cot(θ)= x y , y≠0\cot(\theta)=\frac{x}{y},\:y\ne0 Quadrants: I: csc(θ), sec(θ), cot(θ)>0\csc(\theta),\:\sec(\theta),\:\cot(\theta)>0 II: csc(θ)>0\csc(\theta)>0 sec(θ), cot(θ)<0\sec(\theta),\:\cot(\theta)<0 III:cos(θ), sec(θ)<0\cos(\theta),\:\sec(\theta)<0 cot(θ)>0\cot(\theta)>0 IV: sec(θ)>0\sec(\theta)>0 csc(θ), cot(θ)<0\csc(\theta),\:\cot(\theta)<0 Find the six trigonometric values of 7π 6 .\frac{7\pi}{6}. (Notes)
Radian Measure
What Is Radian Measure and How Is It Related To Degree Measure Definition of Radian Measure Circumference equals 2πr2\pi r A Central Angle is an angle formed by two radii. One radian is the measure of a central angle that intersects an arc length equal to the radius. Formula: θ= s r = arc length radius \theta=\frac{s}{r}=\frac{arc\:length}{radius} Theta is the angle in radians There are about 6.28 radians in a circe, and about 3.14 radians in a semicircle. Comparison of Radian and Degree Measures One revolution: 360 degrees 2π radians2\pi\:radians = 360 degrees 1πr1\pi = 180 degrees Converting Radians to Degrees To convert from radians to degrees, multiply the radian measure by 180 π radians \frac{180}{\pi\:radians} Convert 7π 12 \frac{7\pi}{12} radians to degrees (notes) 1.5 radians ≈90°\approx90^\circ Converting Degrees to Radians To convert from degrees to radians, multiply the degree measure by π radians 180 \frac{\pi\:radians}{180} Convert 120 degrees to radians. (notes) How to Find Arc Length How long is the arc intersected by a central angle of π 4 \frac{\pi}{4} radians in a circle with a radius of 20 inches? (notes) Applications of Arc Length If a clock minute hand is 8 inches long, how far will the tip of the minute hand travel when it moves from 12:00 p.m. to 12:15 p.m.? (notes) The angle measure must be in radians to relate the arc length to the radius.
Graphing Sine and Cosine
Why Do The Graphs Of The Sine And Cosine Functions Have Wave Shapes, And How Can You Change Those Waves? Amplitude: a word for half the distance between the maximum and minimum values of the trigonometric function Period: the horizontal length of one cycle of a periodic function The Sine Function The graph goes in cycles in an order of zero-max-zero-minimum-zero. Periodic Functions Period: The horizontal length of one cycle of a periodic function. The period of the sine function is 2π2\pi The Cosine Function The cycle from the start goes maximum-zero-minimum-zero-maximum. Range is still -1 to 1. Even though the values are different, the period of the cosine function is still 2π2\pi Sine Graph: Min: ( 3π 2 ,−1) Max: ( π 2 ,1)Min:\:(\frac{3\pi}{2},-1)\:Max:\:(\frac{\pi}{2},1) Zeroes: (0,0), (π,0), (2π,0)(0,0),\:(\pi,0),\:(2\pi,0) Domain: All real numbers Range: (−1,1)(-1,1) Cosine Graph: Min:(π,−1)Max: (0,1), (2π,1)Min:(\pi,-1)Max:\:(0,1),\:(2\pi,1) Zeroes: ( π 2 ,0), ( 3π 2 ,0)(\frac{\pi}{2},0),\:(\frac{3\pi}{2},0) Domain: All real numbers Range: (−1,1)(-1,1) Amplitude Amplitude: Half the distance between the maximum and minimum values of the trigonometric functions. ∣a∣= 1 2 (max value−min value)\mida\mid=\frac{1}{2}(\max\:value-\min\:value) If a≠0a\ne0 , then y=asin(x)y=\asin(x) has an amplitude of ∣a∣\mida\mid Vertical Translations The function h(x)=ƒ(x)+k:h(x)=f(x)+k: A vertical translation of f(x) by k units. Make sure you move the entire graph up or down based on that k value. How to Graph a Sine Function Graph y=2.5sin(x)+3y=2.5\sin(x)+3 Graph the parent function, y=sin(x) or y=cos(x)y=\sin(x)\:or\:y=\cos(x) Identify the amplitude, ∣a∣\mida\mid , and stretch or compress, depending on the value of a. If a is negative, reflect over the x-axis. Translate up or down, depending on the value of k.