Download Lesson 17 - Trigonometry

College Algebra to Calculus and the TI-83
Lesson 17
Trigonometry
I. RADIAN AND DEGREE MODES
Degrees: MODE  sets the calculator in degrees mode
Radians: MODE 
sets the calculator in radians mode
To enter seconds, , press ALPHA  (on the + key)
Exercise 1. Convert 251535 to degrees and radians




MODE  ENTER 2nd QUIT (to select Degrees MODE)
25 2nd ANGLE 1 15 2nd ANGLE 2 35 ALPHA  (the + key) ENTER
answer : 25.25972222 degrees
MODE  ENTER 2nd QUIT (to select Radians MODE)
2nd ANS 2nd ANGLE 1 ENTER answer: .4408653209 radians
Exercise 2. Convert 28.6582 to degrees, minutes, seconds.
 MODE 
ENTER 2nd QUIT
 28.6582 2nd ANGLE 4 ENTER
answer: 28 39 29.52
 MODE  ENTER 2nd QUIT
 2nd ANS 2nd ANGLE 1 ENTER
answer: .5001799477
Exercise 3. Find sin(40 28 34) and csc (40 28 34)
(set up the calculator in degrees mode)



MODE  ENTER 2nd QUIT
sin(40 2nd ANGLE 1 28 2nd ANGLE 2 34 ALPHA  ) ENTER
answer.: sin(40 28 34) = .6491309484
x 1
ENTER
answer: csc (40 28 34) = 1.54052121911
Exercise 4: convert 50 degrees to radians and π radians to degrees.




MODE  ENTER 2nd QUIT (to select radian MODE)
50 2nd ANGLE 1 ENTER
answer: .872664626 radians
nd
MODE  ENTER 2 QUIT
π 2nd ANGLE 3 ENTER
answer: 180 degrees
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Exercise 5: convert 2.4 radians and 5/19 radians to DMS
 MODE  ENTER 2nd QUIT
 2.4 2nd ANGLE 3 2nd ANGLE 4 ENTER answer: 137 30  35.535
 (5π19) 2nd ANGLE 3 2nd ANGLE 4 ENTER answer: 47 22  6.316
Exercise 6. Find the trigonometric values of 4815 45
 MODE  ENTER 2nd QUIT
 48 2nd ANGLE 1 15 2nd ANGLE 2 45 ALPHA  STO ALPHA A
 ENTER
 sin(A) ENTER
answer: sin (4815 45)= .7462026301


x
1
cos(A)
x
1
ENTER
ENTER
answer: csc(4815 45)=1.340118568
answer: cos(4815 45)=.6657188857


tan(A)
ENTER
ENTER
answer: sec(4815 45)=1.502135543
answer: tan(4815 45)=1.120897493

x
ENTER
answer: cot(4815 45)=.8921422396
1
II. Graph of the Trigonometric functions –select the radian mode.
Exercise 7. Graph the sine function with amplitude 3 and period 4.
x 

 2 
Y= 3sin 



MODE  ENTER 2nd QUIT
Y=
Y1=3sin(πx2)
WINDOW Xmin=-10 Xmax=10 Ymin=-4 Ymax=4 GRAPH
Exercise 8. Graph the function of exercise 7, with a phase shift of
half a unit to the left.
Y=




 ( x  0.5) 

3sin 
2


and then shift the result 3 units upward.
Y=
Y2=3sin(π(x+0.5)2)
WINDOW Xmin=-10 Xmax=10 Ymin=-4 Ymax=4 GRAPH
Y=
Y3=3sin(π(x+.5)2) +3
WINDOW Xmin=-10 Xmax=10 Ymin=-4 Ymax=8 GRAPH
Exercise 9. Graph the function y=tan(πx), -1≤ x ≤1


Y=
Y1=tan(πx)
WINDOW Xmin=-1
Xmax=1 Ymin=-10 Ymax=10 GRAPH
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Exercise 10. Graph the function y=cot(πx), -1≤ x ≤1. Graph both function in exercises 9 and 10
simultaneously.
 Y=
Y2=1  tan(πx)
 WINDOW Xmin=-1 Xmax=1 Ymin=-10 Ymax=10
 Turn on Y1 and Y2 GRAPH
x
x 
 and Y4= x+sin 
 

 2 
 2 
Exercise 11. Graph the function Y3= xsin 





Y=
Y3=xsin(πx2)
WINDOW Xmin=-10 Xmax=10 Ymin=-20 Ymax=20 GRAPH
Y=
Y4=x + sin(πx  2)
WINDOW Xmin=-5 Xmax=5 Ymin=-6 Ymax=6
Deselect Y3 GRAPH
Exercise 12. Graph the function y  cos x  and y=| cosπx |.
2





Y=
Y1=(cos(πx))^2
WINDOW Xmin=-5 Xmax=5 Ymin=-1 Ymax=2
Y=
Y2= 2nd CATALOG abs ENTER
abs(cosπx)
GRAPH
GRAPH
Exercise 13. Graph the function y=secx and find the equation of the line tangent at x=-1
 Y=
Y3=1  cos(x)
 WINDOW Xmin=-4 Xmax=4 Ymin=-10 Ymax=10
GRAPH
nd
 2 Draw
5 -1 ENTER
answer: y=-2.88248x – 1.03
Exercise 14. Graph the function y=Arccos(x) and evaluate Arccos(-1) Arccos(-0.5)
Arccos(0) Arccos(.5) Arccos(1). Find the equation of the line tangent at x=0.5
 Y=
Y1=2nd cos(x)
 WINDOW Xmin=-2 Xmax=2 Ymin=-2 Ymax=5
GRAPH
 TRACE -1 ENTER
answer: 3.1415927 or π
 -0.5 ENTER
answer: 2.0943951 or 2π/3

0 ENTER
answer: 1.5707963 or π/2
 0.5 ENTER
answer: 1.0471976 or π/3
 1 ENTER
answer: 0
nd
 2 Draw
5 0 .5 ENTER
answer: y=-1.1547 x + 1.6245
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Exercise 15. Graph the function y=Arcsin(x)
Evaluate Arcsin(-1) Arcsin(-0.5) Arcsin(0) Arcsin (0.5) Arcsin(1)
 Y=
Y2=2nd sin(x)
 WINDOW Xmin = -2 Xmax = 2 Ymin = -2 Ymax = 2
GRAPH
 TRACE -1 ENTER
answer: -1.570796 or –π/2
 -0.5 ENTER
answer: -.5235988 or –π/6
 0 ENTER
answer: 0
 0.5 ENTER
answer: .5235988 or or π/6
 1 ENTER
answer: 1.5707963 or π/2
Exercise 16. Graph the function y=Arctan(x)= TAN 1 ( x) . Find the equation of the tangent line at
x=3
 Y=
Y3 = 2nd TAN(x)
 WINDOW Xmin = -20 Xmax = 20 Ymin = -2 Ymax = 2
GRAPH
nd
 2 DRAW 5 3 ENTER answer: y =0.1x +0.949
Exercise 17. Consider the function y = 2sinx+3cosx
a) Graph the function
 Y=
Y4= 2sin(x) + 3cos(x)
 WINDOW Xmin = -4 Xmax = 4 Ymin = -10 Ymax =10
GRAPH
b) Find the amplitude and phase shift of the function
 2nd CALC 4
 move near and to the left of the maximum point ENTER
 move near and to the right of the maximum point ENTER
 move to a point between the left bound and the right bound ENTER
2
2
maximum occurs at x = 0.5880011 and y=amplitude=3.6055513 = 2  3
c) Use the fact that the function is a cosine function with amplitude 3.6055513 and shifted
0.5880011 to the right, to write the function as a cosine function and graph the result.
Y= 3.6055513cos(x-0.5880011)
 Y= Y5=3.6055513cos(x -0 .5880011) GRAPH (observe that the graphs are the same)
Exercise 18: Let f (x )  (sin 3x)
2
a) Graph function

Y=

WINDOW : Xmin = -3
Y1=(sin(3x))^2
Xmax = 3
Ymin = -2 Ymax = 2 GRAPH
2nd QUIT
b) Find the value of f(1.3)

GRAPH
TRACE
1.3
ENTER
answer: 0.47302229
c) Find the equation of the tangent line at x=2

2nd DRAW 5
2 ENTER
answer: y = -1.609709095708x + 3.29 2nd DRAW 1
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Exercise 19. Graph the function y=xsin(x). Find a) the slope at x=3, b) the equation of the
tangent line at x=-5, c) the equation of the tangent line at x=4.5 d) the maximum value of the
function for 0 < x < 5).

MODE (select Radian mode) 2nd QUIT

Y=

WINDOW: Xmin = -10

2nd CALC

2nd DRAW 5

2nd DRAW 1 (to clear the tangent line)

2nd DRAW 5

2nd DRAW 1

MATH 7

VARS Y-VARS 1

Y3( 2nd ANS )
Y3 = xsin(x) 2nd QUIT
6
Xmax = 10
3 ENTER
-5 ENTER
Ymin = -10 Ymax = 10 GRAPH
answer: slope = dy/dx = -2.828857
answer: y=-.4593868957x-7.0915
4.5 ENTER
answer: y=-1.9261105682x+4.2686
fMax(Y3, x, 0, 5) ENTER
x = 2.028759264
3
answer: 1.819705741
Exercise 20. Use the laws of sine and cosine to solve the following triangle.
B
2
5
A
2
2
16




2
2
a c  b
b  a  c  2ac cos(B) or cos(B) 
2ac
a
b
c


13
sin(A) sin(B) sin(C)
2
C
MODE  ENTER 2nd QUIT
2nd cos( (13^2 +5^2 – 16^2) (2513))
2nd sin( sin(2nd ANS) 1316) ENTER
2nd sin( sin(2nd ANS)  513) ENTER
answer: B = 118.48 degrees
answer: A = 45.57 degrees
answer: C = 15.94 degrees
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