Similar Triangles, Right Triangles, And The Definition Of The Sine, Cosine And Tangent Functions Of Angles Of A Right Triangle Worksheets With Answer Key Page 24
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43+
=
2
2
2
x
y
r
so
2
2
x
y
+
=
. 1
r
r
Therefore, by the definitions of the sine and cosine functions,
θ
θ
+
=
2
2
cos
sin
1
.
Similarly,
θ
θ
+
=
2
2
1
tan
sec
and
θ
θ
+
=
2
2
cot
1
csc
.
These identities are called the Pythagorean identities.
The reciprocal, quotient and Pythagorean identities are used frequently. Memorize them or learn to
construct them quickly by reasoning from the definitions as in the discussion above.
Problems:
Memorize the basic identities above. Then, without referring to the statements of the basic identities,
fill in the blanks to form one of the reciprocal, quotient or Pythagorean identities, or one of their
alternate forms.
θ
cos
=
θ
+ 1
=
2
89. _____________________________
88.
tan
________________________
θ
sin
1
=
θ
θ
+
=
2
2
91. _____________________________
90.
____________________
sin
cos
θ
cos
θ
sin
=
θ
=
2
92. ___________________________
93.
csc
___________________________
θ
cos
1
=
θ
=
94. ___________________________
2
95.
cos
___________________________
θ
tan
θ
θ
θ
θ
=
=
−
2
2
97.
________________________
tan
cot
96. ____________________
csc
cot
1
=
θ
θ
−
=
98. ___________________________
2
2
99.
tan
sec
_____________________
θ
cot
θ
θ
θ
=
=
1 −
2
101.
_______________________
sin
cot
100. ______________________
cos
1
θ
=
=
103. ___________________________
102. _________________________
csc
θ
sec
1
θ
θ
=
=
105. ___________________________
104.
cos
tan
______________________
θ
csc
Given the quadrant of an angle and the value of one of its trigonometric functions, we can use the
preceding identities to find the values of the other trigonometric functions.
θ
θ
Example: Find cot
if tan
= -5.
1
1
1
θ
θ
=
=
=
−
Solution: Since
cot
,
we
have
cot
.
θ
tan
5 -
5
- 24 -
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