Basic Trigonometric Identities
T, 3.1: Students prove that this identity is equivalent to the
Pythagorean theorem (i.e., students can prove this identity by
using the Pythagorean theorem and, conversely, they can prove
the Pythagorean theorem as a consequence of this identity).
T,3.2: Students prove other trigonometric identities and simplify
others by using the identity cos2 (x) + sin2 (x) = 1. For example,
students use this identity to prove that sec2 (x) = tan2 (x) + 1.
Basic Trigonometric Identities
Objectives
• Derive the Pythagorean
Trigonometry Identity
• Identify and use reciprocal
identities, quotient
identities, Pythagorean
identities, symmetry
identities, and oppositeangle identities.
Key Words
• Identity
• Trigonometric Identity
• Reciprocal Identities
• Quotient Identity
• Pythagorean Identities
• Symmetry Identities
• Opposite-Angle Identities
Deriving the Pythagorean
Trigonometric Identity
•
•
•
•
•
•
Pick a coordinate point on the unit
circle.
Draw the appropriate triangle ABC.
Write the Pythagorean theorem
for this triangle with sides a, b, and
c.
Divide by c-squared.
Replace the appropriate values
with respect to the trigonometric
function sine, and cosine.
Finish with the last touch ups.
Study this exact proof. You will be
tested on it. The exact same question
and solution.
Example 1

Suppose x = .
3
Prove that sec x cot x = sin x is not a
trigonometric identity by producing a
counterexample.
?
sec x cot x  sin x

 ?

sec cot  sin
3
3
3

 ?
 
3

2 
3
2
 3 ?
(2) 
 
 3 
3
2
 1 
 1 


 2 





1
2
2
3
3
≠
3
2

Replace x with .
3
Reciprocal Identities
Quotient Identities
Pythagorean Identities
6
a. If cot  = , find tan  .
5
1
Example 2
tan  =
Choose an identity that involves tan  and cot .
cot 
1
5
6
=
or
Substitute
for cot  and evaluate.
Use the given information to find the
6
6
5
trigonometric value.
5
a.
b.
If cot  =6/5 , find tan .
If sec  =5/4 , find cot .
5
b. If sec  = , find cot  .
4
Since there are no identities relating sec  and cot , we must use two identities,
one relating sec  and tan  and another relating cot  and tan .
sec2  = 1 + tan2 
 
5
4
2
= 1 + tan2 
Pythagorean identity
5
Substitute for sec .
4
25
=1 + tan2 
16
9
= tan2 
16
3
 = tan 
4
Now find cot .
1
tan 
4

3
cot  =
Reciprocal identity
02-15-2012
Derive the Pythagorean Trigonometric
Identity (10 Minutes)
• Draw the appropriate triangle ABC.
• Write the Pythagorean theorem for this
triangle with sides a, b, and c.
• Divide by c-squared.
• Apply appropriate exponential property
• Replace the appropriate values with respect to
the trigonometric function sine, and cosine.
• Finish with the last touch ups.
Symmetry Diagram
Symmetry Identities
Example 3
Express each value as a trigonometric
function of an angle in Quadrant I.
a. sin 765°
a. sin 765°
b. sin (-19π/3)
765° and 45° differ by a multiple
c. cos 935°
765° = 45° + 2(360°)
d. cot 11π/4
sin 765° = sin (45° + 2(360°))
= sin 45°
of 360°.
Case 1, with A = 45° and k = 2
19
b. sin ()
3
19

The sum of and is a multiple of 2.
3
3
19

- 3 = -6 - 3
19


sin () = sin (-6 - )
Case 3, with A = and k = -3
3
3
3

= - sin 3
Example 3
Express each value as a trigonometric
function of an angle in Quadrant I.
a. sin 765°
c. cos 935°
Relate 935°
b. sin (-19π/3)
935° = 35° + 5(180°)
c. cos 935°
cos 935° = cos (35° + 5(180°))
d. cot 11π/4
= - cos 35°
d. cot
935° and 35° differ by an odd multiple of 180°.
Case 2, with A = 35° and k = 3
11
4
11
12

and , which is
or 3, is an odd multiple of .
4
4
4


= 3 - 4
Case 4 with A = 4 and k = 2
11
cos
4
=
Rewrite using the quotient identity.
11
sin 4

cos 3 - 4 
=

sin 3 - 4 

- cos
4

=
or - cot 4
Quotient identity

sin 4
The sum of
11
4
11
cot 4
Opposite-Angle Identities
Example 4
Simplify cos x cot x + sin x.
cos x
cos x cot x - sin x = cos x  sin x + sin x
cos2 x
=
+ sin x
sin x
1 - sin2 x
=
+ sin x
sin x
1
= sin x - sin x + sin x
1
=
or csc x
sin x
Definition of cot x
Pythagorean identity: sin2 x + cos2 x = 1
Reciprocal identity
Example 5
PROBLEM SOLVING In the Cartesian coordinate
system, points are expressed as ordered pairs of their
x- and y-coordinates, (x, y), on a coordinate plane
where x is the signed horizontal distance from the
origin and y is the signed vertical distance from the
origin. Another way of expressing a point P is in
polar coordinates, (r, ), where r is the signed
distance of a point from a fixed origin O, or pole,
and  is the signed angle from the initial ray to the
ray OP. Using polar coordinates, the equation
ke
r=
, where 0 < e < 1, represents an ellipse
1 + e cos 
F fM A X
that has a focus at the origin and directrix x = k to a. μS = F N
the right of the origin. Show that the equation
mg sin 
μS =
ke sin  csc2 
mg cos 
r=
represents the same ellipse.
csc  + e cot 
μS = tan 
b. μS = tan 60°
μS = 3
μS  1.732050808
The coefficient of static friction is about 1.73.
Conclusions
Summary
• Write an expression that
contains all six
trigonometric functions and
is equal to 3.
sin2x+cos2x+sec2x-tan2x+csc2x-cot2x
Assignment
• Pg427 #(18-43 ODD, 44-53
ALL)