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6-3 Properties of Trigonometric Functions

Pythagorean Identities

Find the exact value of the remaining trig functions

tan A = -1/3, sin A>0

1

By Pythagorean theorem: x²+y²=1

y

x

Since x= cosine & y= sine, we can say: cos² + sin² = 1

sin A = 1/( 10 ) csc A= 10

( 10 )/10

cos A= -3/( 10 ) sec A= -( 10 )/3

= (-3 10 )/10 cot A= -3

You have five minutes to do these problems as they might be on the final!

A Real World Example

To get the other identities:

Find the exact value of this expression:

sin (- /12) csc (25 /12)

Divide cos² + sin² = 1 by sin²

Page 393 in the textbook:

53) sec = 2

75) sec (- /6)

83) cos 400 sec 40

85) sin (- /12) csc (25 /12)

You get: cot² + 1 = csc²

Divide cos² + sin² = 1 by cos²

Video Explanation

a) 1

b) -1

c) 0

75) sec (- /6)

.

83) cos 400 sec 40

.

53) sec = 2, sin <0

85) sin (- /12) csc (25 /12)

You get: 1 + tan² = sec²

180

.

cos = 1/2

sin = - 3

-30

.

(1/ (cos 40 ))

=

cos (360 + 40 )

=

.

sin (-15 ) csc (360 +15 )

=

1

6

.

cos 40

=

(1/ (cos 40 ))

.

- sin (15 )

( 1/ (sin 15 ))

2

=

1

=

sec (-30 )= sec (30 )

-

3

2

-1

=

csc = -2

3

sec (30 )= 1/ cos (30 )

3

tan = -

3

1/ cos ( 3 / 2 )= 2

3

cot = -

3

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The Correct Answer