Fundamental Trigonometric Identities

Contributed by:

Sharp Tutor Mon, Jan 17, 2022 08:59 PM UTC

This presentation is about some fundamental trigonometric identities along with their examples discussed.

1. Do Now

2. Chapter 5
Trigonometric Identities
Section 5.1
Fundamental Identities
SWBAT use the fundamental identities of
Trigonometry functions to find trigonometric functions
values given one value and the quadrant.

3. Trigonometric Identities
Quotient Identities: (write these down on an index card)
sin  cos 
tan   cot  
cos  sin 
Reciprocal Identities:
1 1 1
sin   cos   tan  
csc  sec  cot 
1 1 1
cot   sec  csc 
tan  cos sin 

4. Trigonometric Identities Cntd.
(write these down on an index card)
Negative-Angle Identities:
sin(   )  sin  cos(   ) cos  tan(   )  tan 
csc(  )  csc sec(  ) sec cot(  )  cot 

5. Where did our Pythagorean Identities come from?
Do you remember the Unit Circle?
What is the equation for the unit circle?
x 2 + y2 = 1
What does x = ? What does y = ?
(in terms of trig functions)
Sin2 θ + cos2 θ = 1
Pythagorean
Identity!

6. Take the Pythagorean Identity and
discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ = 1 .
cos2θ cos2θ cos2θ
tan2θ + 1 = sec2θ
Quotient Reciprocal
Identity another Identity
Pythagorean
Identity

7. Take the Pythagorean Identity and
discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ = 1 .
sin2θ sin2θ sin2θ
1 + cot2θ = csc2θ
Quotient Reciprocal
Identity a third Identity
Pythagorean
Identity

8. Using the identities you now know, find the trig
5 value :
If tan   and  is in quadrant II, find sec .
3
tan 2  1 sec 2 
2
Look for an identity  5 2
   1 sec 
that relates tangent and 3
secant. 25
1 sec 2 
9
tan 2   1 sec 2  34
sec 2 
9
34
sec  
9
34
sec  
3

9. Using the identities you now know, find the trig
value :
If tan   5 and  is in quadrant II, find cot ()
3
1
cot(  ) 
tan(  )
1
cot(  ) 
 tan 
1 3
cot(  )  
   3 5
5

10. Using the identities you now know,
find the trig value.
1.) If cosθ = 3/5, find csc θ.
sin 2   cos 2  1
2
3 
sin 2     1
5 
25 9
sin 2   
25 25
2 16
sin  
25
4
sin  
5
1 1 5
csc    
sin  4 4
5

11. Using the identities you now know, find
the trig value.
2.) If cosθ = 3/4, find secθ
1 1 4
sec    
cos  3 3
4

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