6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
6.3 Trig Functions of ANY angle
Now you will be utilizing the entire coordinate axes (full circle) so trig functions can have negative values.
Ex1: Let (3, ­4) be a point on the terminal side of an angle. State the 6 trig ratios of the angle.
Steps to find trig ratios of any angle given a coordinate
1) Draw the point for the given ordered pair.
2) Draw a right triangle. (this will always be drawn to the x­axis)
3) Label the sides appropriately with + or ­ and values.
4) Set up the trig ratios.
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
In general, the trig ratios can be defined terms of x, y, and r (radius) rather than adj, opp, and hyp.
sin
csc
cos
sec
tan
cot
This may be useful if you don't want to draw the out the triangle and you are given an ordered pair. (?) You don have to memorize something else though..........
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
You can also use the formulas just given in order to determine a quadrant where an angle terminates when given general information........
For example: If sin A <0 , in which quadrant(s)
would A lie?
Ex: If cos B >0 and csc B < 0, in which quadrant(s) would B lie?
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
There is an "easy" way to remember the signs of the trig functions to help with these questions.
(you will also want to learn this for later use)
So let's try it on a HARD example.
Ex: If csc M <0 and cot M<0, in which quadrant(s) would M lie?
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
Quadrant Angles: Angles whose terminal side is either on the
x or y axis.
**These trig ratio values are usually 1, 0, or undefined**
A strategy to evaluate a quadrant angle value is to locate the angle in question and use a radius of 1. Write the ordered pair for the angle. Set up the trig ratio using the coordinate formulas with x,y,r.
EX 1: Evaluate Ex: Given cos B = and B lies in Quad III
find tan B. (the book asks for all 6 ratios)
How to find trig ratios given quadrant & 1 ratio:
1) Look and see whether x & y should be positive or negative
2) Determine which quadrant(s) would satisfy #1
3) Sketch a triangle in the correct quadrant
4) Label the triangle with given information
5) Find missing side length
6) Set up trig ratios
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
EX: Given csc M= and cos M<0
find the 5 remaining trig ratios.
Assignment:
p.474 #1,7, 11­14, 15, 17,29­36
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
"HARD" EX: The terminal side of the angle lies on the line 4x + 3y=0.
The angle lies in quadrant IV. State the values of the six trig ratios.
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
Reference Angle: An angle that measures between 0 and 90.
This is the angle that will be used for in the triangles drawn yesterday.
Strategy for finding reference angles:
Sketch the angle. Draw the triangle to the x­axis.
Determine the acute measure of the angle by the origin.
(Typically subtract from 180 or 360....)
EX3: State the reference angle.
Assignment:
p. 474 #19, 21, 23, 27
37­42 all
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
Answers:
Now we will try to combine the concepts of finding exact trig ratios with reference angles.....
EX: Find the exact value of cos 300o
Finding exact trig values for ALL angles
STEPS:
1. Sketch the given angle.
2. Determine the reference angle. Label it.
3. Label the sides of the triangle with + and ­ values
4. Set up the requested trig ratio.
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
Ex: Find the exact value of Finding angles for 0< <360 given a trig ratio.
1. Determine the quadrant(s) the angle could lie in (based on +/­).
2. Sketch the right triangle (to the x­axis)
3. Label the sides according to the trig ratio.
4. Use the reference angle (and x­axis) to find the actual FULL
angle of revolution. (add or subtract 0o, 180o, 360o, etc)
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
You just found angles given exact trig ratios using special triangles. We can find other angles of approximate trig ratio values using a scientific calculator.
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
Assignment:
p.475 #69­74 all, 75, 77
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
Due to heat schedule, your 6.1­6.3 exam is moved to this Wednesday.
Work in groups to complete the study guide.
Study guides are always optional in my class but HIGHLY RECOMMENDED!!!
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
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6.3 trig ratios of any angle, reference angles.notebook
September 10, 2014
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