View
222
Download
2
Category
Preview:
Citation preview
Trigonometric Functions
The Unit Circle
The Unit Circle Definition: A circle whose center is the origin and whose radius
has a length of one.
Based on the definition, give the coordinates for the x- and y-intercepts for the diagram below.
(__, __)
(__, __)
(__, __)
(__, __)
1 , 0
0 , 1
-1 , 0
0 , -1
The Unit Circle Let’s determine the coordinates on the unit circle for a 45 angle.
2h l
45
(__, __)
Drop the perpendicular from the pointon the circle to the positive x-axis.
What type of triangle is created?An isosceles right triangle
What is the length of the hypotenuse? Why?
1 because it is a unit circle
1
What is the relationship between thehypotenuse and a leg of an isosceles right triangle?
What is the length of each leg?1 2
22l
What is the x-coordinate of the point? 2
22
2The y-coordinate?
2
2
2
2
2
2
2
2
The Unit Circle
2
2
2
2 2
2
135
(__, __)
What would be the coordinates of the point on the circle if we draw a 135 angle? Explain.
What would be the coordinates of the point on the circle if we draw a 225 angle? Explain.
2
2(__, __)
225
The Unit Circle
2
2
What would be the coordinates of the point on the circle if we draw a 315 angle? Explain.
315
(__, __)2
2
The same procedure can be used to findthe coordinates for a 30 angle and a 60 angle. Visiting all quadrants would
result in the following figure.
The Trigonometric Functions Let t be the measure of a central angle and let (x, y) be the point on the unit
circle corresponding to t. The following are the definitions for the six trigonometric functions based on the unit circle.
sin t y 1csc t
y
cos t x 1sec t
x
tany
tx
cotx
ty
The Trigonometric Functions
1sin
2t y
6t
3 1,
2 2
Step 1: Identify the quadrant the terminal side of the angle
is located. Step 2: Identify the coordinates that correspond with that angle.
How to determine the trigonometric values for a given angle
Ex. 1: Evaluate the six trigonometric functions for
Quadrant I
Step 3: Follow the definitions
3cos
2t x 1
csc 2ty
1 2 2 3sec
33tx
32
12
cot 3x
ty
12
32
1 3tan
33
ytx
The Trigonometric Functions
5
4t
2 2,
2 2
2sin
2t y
Ex. 2: Evaluate the six trigonometric functions for
Step 1: Identify the quadrant the terminal side of the angle is located.
Quadrant II
Step 2: Identify the coordinates that correspond with that angle.
Step 3: Follow the definitions
1 2csc 2
2ty
2cos
2t x
22
22
tan 1y
tx
1 2sec 2
2tx
22
22
cot 1x
ty
The Trigonometric Functions
7
3t
7 7 62
3 3 3
1 3,
2 2
Ex. 2: Evaluate the six trigonometric functions for
In this problem the value of the angle exceeds 2. Find the coterminal anglewhose value lies between 0 and 2
Step 1: Identify the quadrant the terminal side of the angle is located.
Quadrant I
Step 2: Identify the coordinates that correspond with that angle.
Step 3: Follow the definitions
1cos
2t x
3sin
2t y tan 3
ytx
1 2 3csc
3ty
1sec 2t
x
3cot
3
xty
The Trigonometric Functions Recapping for evaluating the six trigonometric
functions for any given angle. If the given angle does not lie between 0 and 2, find its
simplest positve coterminal angle and use that value. Identify the quadrant the terminal side for the given angle lies. Determine the coordinates on the unit circle for the given
angle. Follow the definitions for the six trigonometric functions.
Even and Odd Trig Functions What is an even function?
An even function is when one substitutes a negative value into the original function and the outcome is the same as its positive value.
f (- x) = f (x)
Ex. f (x) = x2
f (2) = 22 = 4
f (-2) = (-2)2 =4
Therefore, f (-2) = f (2) and f (x) = x2 is an even function.
Even and Odd Trig Functions What is an odd function?
An odd function is when one substitutes a negative value into the original function and the outcome is the opposite of its positive value.
f (- x) = - f (x)
Ex. f (x) = x3
f (2) = 23 = 8
f (-2) = (-2)3 = - 8
Therefore, f (-2) = - f (2) and f (x) = x3 is an odd function.
Even and Odd Functions Let’s determine whether or not the sine function is even or odd.
f(x) = sin x
What is the sin 30? What is the sin (- 30)? sin (- 30) = - sin (30) sin (- 30) = -( ½ ) = - ½
Because the sin (- 30) = - sin (30), the sine function is odd.
sin (-x) = - sin x
½ - ½
Even and Odd Trig Functions Lets determine whether the cosine function is even or odd.
f (x) = cos x
What is the cos 60? What is the cos (- 60)? cos (- 60) = cos 60 cos (- 60) = ½
Because the cos (- 60) equals cos 60, cosine is an even function
cos (- x) = cos x
½½
Even and Odd Trig Functions Determine what the other 4 trig functions are – even or odd
tan (x)
csc (x)
sec (x)
cot (x)
odd function
odd function
odd function
even function
Evaluating Trig Fuctions with a Calculator
Steps Set the calculator into the correct mode
Casio From the MENU select RUN Press SHIFT, then MENU (above MENU you should see SET
UP) Scroll down and highlight Angle Select F1 for Deg, Select F2 for Rad Press EXE (Blue key at bottom of calculator) Type the measure of the angle into the calculator Press EXE
Evaluating Trig Fuctions with a Calculator
Using the calculator find the values for the following (round to 4 decimal places): 1. sin 214 2. tan (-175 ) 3. cos 5/9 4. sin 14/5
-0.5592
0.0875
-0.1736
0.5878
Evaluating Trig Fuctions with a Calculator
To evaluate cosecant, secant, or cotangent on the calculator, follow these steps:
Casio Make sure you are in the right mode (degree or radian) In parentheses type the trig function on the calculator that is the
reciprocal function of the one being evaluated. On the outside of the parentheses, press SHIFT, then ). A - 1 should
appear next to the parentheses. Example: Evaluate csc 40
Make sure you are in the degree mode. Type into the calculator (sin 40) Press SHIFT, then ). Your screen should now read (sin 40) - 1
Press EXE. Your answer should be 1.5557
Evaluating Trig Fuctions with a Calculator
Using the calculator find the values for the following (round to 4 decimal places): 1. sec 297 2. cot 19/12 3. csc (- 11.78) 4. sec /2
2.2027
-0.2679
1.4128
Ma ERROR - WHY?
4.2 Trigonometric Fuctions:
The Unit Circle Can you
Sketch the unit circle and place key angles and coordinates on it?
Explain how these coordinates are derived using either 30, 45 , or 60 ?
Determine the trig value for certain angles using the unit circle?
Explain why a trig function is either even or odd? Evaluate a trig function using the calculator?
Recommended