Introduction to Trigonometry Algebra 2
Angle Rotation Angle formed by the starting and ending positions of a ray that rotates about its endpoint Use θ to represent the angle measure Greek letter theta Counterclockwise rotations give positive θ, angle measures Clockwise rotations give negative θ, angle measures https://www.desmos.com/calculator/6omeu5xv 3c
Definitions Standard Position initial side of the angle that lies on the positive x- axis Terminal Side ending position of ray terminates Coterminal Angles Angles that share the same terminal side
New Ways to Measure Angles In Geometry, measured angles in degrees In Trigonometry, often measure angles in radians Dealing with angle rotations in trig, which means circles What special number do we associate with circles? Degrees don t contain pi, radians do (more accurate measure)
Conversions Convert degrees to radians π 180 Convert radians to degrees 180 π
Examples 20 π 8 315 4π 3-60 7π 12-540 13π 6
Arc Length Arc Length- part of the circumference Formula s = rθ s is the arc length (how far the point traveled in a circle) r is the radius θ is the measure of the central angle (angle that forms the arc); also known as angular velocity
Example A plane is flying in a circular pattern with a radius of 45 miles. How many miles does the plane fly on the circular path over 140 degrees?
Example A seat on a Ferris wheel travels one quarter of a revolution. The length of the arc traveled by the seat is 5π feet. Find the radius of the Ferris wheel.
Example A point on the Sun s equator makes a full revolution once every 25.38 days. The Sun has a radius of about 432,200 miles at its equator. What is the angular velocity in radians per hour of a point on the Sun s equator? What distance around the Sun s axis does the point travel in 1 hour?
Example A neutron star (an incredibly dense collapsed star) in the Sagittarius Galaxy has a radius of 10 miles and completes a full revolution every 0.0014 seconds. Find the angular velocity of the star in radians per second, then use this velocity to determine how far a point on the equator of the star travels each second.
Homework Pg. 833: 7 16
Bell Ringer Convert the following from either degrees to radians or vice versa 147 9π 5
Basic Trigonometric Functions Sine Cosine Tangent
Mnemonics Soh Cah Toa In the order Sine, Cosine, Tangent Oscar Had A Hold On Arthur Oscar Had A Heap Of Apples
Special Right Triangles
Hand Trick Pinky is the x-axis Thumb is the y-axis Ring Finger is 30 Middle Finger is 45 Pointer Finger is 60
Hand Trick Sine is bottom over 2 Cosine is top over 2 Tangent is bottom over top
Quadrant I II III IV θ 45 sin θ cos θ tan θ
Quadrant I II III IV θ 30 sin θ cos θ tan θ
Quadrant I II III IV θ 60 sin θ cos θ tan θ
Signs of Each Trig Function in Each Quadrant
Examples cos 16π 3 cos 1260 tan 11π 4 sin 600 tan 5π 6 sin 5π 6
Homework Pg. 847: 2 9 (together) Worksheet
Introduction to Trigonometry: Day 2 Algebra 2
Bell Ringer Convert the following from either degrees to radians or vice versa 147 9π 5
Basic Trigonometric Functions Sine Cosine Tangent
Mnemonics Soh Cah Toa In the order Sine, Cosine, Tangent Oscar Had A Hold On Arthur Oscar Had A Heap Of Apples
Special Right Triangles
Review Quadrants
Who s Positive?
Trigonometry Tricks Algebra 2
Take a paper plate, one piece of string, and 4 different colored markers or pens
Draw an axis on your plate with one of your colors With the same color, write the coordinate points that are on the axis With a new color, write the degree measures With your 3 rd color, write the radian measures With your remaining color, write the letters and quadrant numbers
On the Back Write both special triangles
Hand Trick Pinky is the x-axis Thumb is the y-axis Ring Finger is 30 Middle Finger is 45 Pointer Finger is 60
Hand Trick Sine is bottom over 2 Cosine is top over 2 Tangent is bottom over top
Examples cos 16π 3 cos 1260 tan 11π 4 sin 600 tan 5π 6 sin 5π 6
Homework Worksheet
Graphing Trig Functions Algebra 2
Review sin 270 cos 0 cos 930 sin 765 cos 3π tan 630 cos 17π 3 sin 5π 4
Trig Graphs Sine https://www.desmos.com/calculator/uoytpimgyy Cosine https://www.desmos.com/calculator/61lo0ylp59 Tangent https://www.desmos.com/calculator/lrtl0cv3ob
Definitions Crest/Trough: Max/Min Midline: equation of the line that is halfway between the min and max Amplitude: distance between min and midline, OR distance between max and midline Period: distance it takes to complete one full cycle Frequency: how many cycles it completes in a certain interval Phase Shift: horizontal translations Vertical Shift: vertical translations
Standard Form of Sine & Cosine y = a sin bx + c + d y = a cos bx + c + d Amplitude: a Midline: y = d Period: 2π b OR 360 b Phase Shift: c b Vertical Shift: d
Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 6 cos 4θ + 300 2
Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 1 + 8 sin(5θ + 60)
Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 2 sin 8θ + π 3 + 3
Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 6 cos θ 3 + 5π 6 2
Standard Form of Tangent y = a tan bx + c + d Amplitude: none Midline: y = d Period: π b OR 180 b Phase Shift: c b Vertical Shift: d
Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 1 tan 2θ + 60 + 4 3
Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 5 tan 6θ 120 1
Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 1 6 tan θ 4 + 5π 6 + 1
Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 8 tan θ 5 + 3π 2 + 1
Pythagorean Identity Algebra 2
Pythagorean Identities When doing trigonometry, what 2 common shapes are we working with? What is another formula we associate with triangles?
Example Given that sin θ = 0.766 where 0 < θ < π, find cos θ. 2
Example Given that cos θ = 0.906 where π < θ < 3π 2, find sin θ.
Example Given that tan θ = 2.327 where π 2 values of sin θ and cos θ. < θ < π, find the
Example Given that tan θ = 4.366 where 3π 2 values of sin θ and cos θ. < θ < 2π, find the
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