Section 7 4 Evaluating and Graphing Sine and
- Slides: 54
Section 7 -4 Evaluating and Graphing Sine and Cosine Objective: Day 1: Reference angles. Day 2: Parent Graphs of sine and cosine function Day 3: UC and parent graphs; application problems.
The curve bank! HTTP: //CURVEBANK. CALSTAT ELA. EDU/UNIT. HTM
The site below demonstrates reference angles. • Reference Angles
Evaluating Sine SO H H and Cosine A C First we must Understand the Reference Angles
What is a Reference Angle? Termin al Side 55 o Reference Angle 125 o A reference angle is the positive acute angle formed by the terminal side of the angle and the x-axis.
Finding the Reference Angle Quadrant I In the first quadrant the angle is its own reference angle!
Quadrant II In the 2 nd quadrant, the reference angle is the angle formed by the terminal side of the 120 o angle and the xaxis. To find the reference angle measure: 180 o – 120 o. 180 o – 1200 = 60 o
Quadrant III In the 3 rd quadrant, the reference angle is the angle formed by the terminal side of the 240 o angle and the x-axis. To find the reference angle measure: 240 o – 180 o. 240 o – 1800 = 60 o
Quadrant IV In the 4 th quadrant, the reference angle is the angle formed by the terminal side of the 315 o angle and the x-axis. To find the reference angle measure: 360 o – 315 o.
Reference angles •
Reference Angles •
Remember: The reference angle is measured from the terminal side of the original angle "to" the x-axis (not the y -axis).
More on Reference angles • Consider angle 30°. • Draw the line segments that represent sin 30° and cos 30°.
Can you find angle whose terminal ray is in the 2 nd quadrant such that : • • • The reference angle is 30° How does sin 30° = compare to sin 150° ? How does cos 150° compares to cos 30° ?
Can you find angle whose terminal ray is in 3 rd quadrant such that : 1. 30° is its reference angle. 2. How does sin 210° compare to sin 30° ? 3. How does cos 210° compare to cos 30° ?
4 th quadrant: Find a 4 th quadrant angle for which: 1. 30° is its reference angle. 2. How does sin 330° compare to sin 30° ? 3. How does cos 330° compare to cos 30° ?
The important part of reference angles •
A graphic visual of reference angles
What if the angle is bigger than 360 o or 2π? Find an angle between 0 o and 360 o that is co-terminal. o o o 695 – 360 = 335 To find the reference angle for 695 o Find the reference angle for 335 o : 360 o – 335 o = 25 o
Reference Angles • Express each in terms of a reference angle: • sin 695° • cos 124° • sin -190°
Reference Angles • sin 25°= 0. 4226 • Without using a calculator find the following: • • • sin 155° sin 205° sin 335 ° sin -25 ° sin 515 °
Reference Angles • Sin 25°= 0. 4226 • Use your calc to find: Cos 25°. • Without using a calculator find the following: • Cos 155° • Cos 205° • Cos 335 °
Sec 7. 4 Day 2 • Review: express each in terms of the reference angle: sin 473° cos -123 °
The unit circle • Now that we understand reference angles we can build the unit circle. • We will determine the key values of 1 st quadrant angles, and then use reference angles to determine the key values of the 2 nd, 3 rd, and 4 th quadrants.
Sine and Cosine of Special Angles 45 o-90 o 45 o 30 o-60 o-90 o
Notice a pattern? Ѳ degrees Ѳ radians 0 30 0 Sin Ѳ Cos V 0 1 1 0 45 60 90
Find the exact value of each: is in the 3 rd Quadrant Sin is negative in the 3 rd Quadrant
Graphing sine and cosine functions Graphing using your calculator. • When angle measure is in degrees or in radians. Graphing without your calculator. • When angle measure is in degrees or in radians.
Graphing with the 5 key points • 1 complete period of Sine or Cosine can be graphed using the 5 key points. • For each specific equation, the horizontal spacing between each key point is constant. i. e. if it is 3 units between point 2 to point 3, then it is also 3 units between point 4 and point 5. • For both sine and cosine, the 5 key points will always be; maximum values, minimum values, and points on the axis of the wave. (the middle) • the axis of the wave for the parent graphs is the X-axis.
Critical values of the parent graph of the sine function: Radians The Period The amplitude The coordinates of the starting point aka Y-intercept aka The maximum First x intercept The minimum point Second x intercept End point Degrees Notes
Critical values of the parent graph of the sine function: Radian Degree Notes s s The Period 2π 360 The amplitude 1 1 The coordinates of the starting point aka Y-intercept (0, 0) Sine “starts” in the middle and increases) key point #1 The maximum (90, 1) Key point #2 Second x intercept (180, 0) Key point #3 The minimum point (270, -1) Key point # 4 End point (3 rd x- (360, 0) Key point #5
Critical values of the parent graph of the cosine function: Radians The Period The amplitude The coordinates of the starting point aka Y-intercept aka The maximum First x intercept The minimum point Second x intercept End point Degrees Notes
Critical values of the parent graph of the cosine function: Radian Degree Notes s s The Period 2π 360 The amplitude 1 1 The coordinates of the starting point aka Y-intercept Aka the maximum (0, 1) cosine “starts” at the maximum and decreases key point #1 The first x-intercept (90, 0) Key point #2 The minimum point (180, -1) Key point #3 The second x intercept (270, 0) Key point # 4 End point (back to max) (360, 1) Key point #5
All at once! What do you think: a) The coordinates of the intersection point are? b) Where would you find the intersection points on the UC?
All at once but more than once!
See the site below for cool demonstartion SIMULATION OF SINE AND COSINE GRAPHS
How to use your calculator to find sin and cos • Before doing any calculations involving trig functions always check the calculator mode.
Make sure to check the mode then evaluate the expressions below: • Find the value of each expression to three decimal places. • • A. ) sin 122° B. ) cos 237° C. ) cos 5 D. ) sin (-2)
Latitude • The latitude of a point on Earth is the degree measure of the shortest arc from that point to the equator. For example, the latitude of point P in the diagram equals the degree measure of arc PE.
How far is Rome (aka Roma) from the equator? • The Latitude of Rome is approximately 42 N. The radius of earth is approximately 3963 miles Remember s=r where is measured in radians.
How far is Santiago, Chile from the equator? • The latitude of Santiago is 33º 28´ S.
Homework written exercises sec 7. 4 Part 1: ü#1 -17 odds ü#21 -24 ALL • Use your 4 -day weekend wisely. Part 2: #26 -31 All
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