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 26
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433
4
θ
θ
θ
=
−
=
−
111. Find
cot
if
sin
and
cos
.
5
5
7
θ
θ
θ
=
112. Find
sin
if
sec
and
is
a
quadrant
IV
angle.
4
θ
θ
θ
=
113. Find
sec
if
csc
2
and
is
a
quadrant
II
angle.
Any trigonometric expression can be written entirely in terms of the sine and cosine functions by using
the reciprocal and quotient identities.
(
)
θ
sin
θ
θ
θ
tan
sin
cos
θ
θ
=
=
⋅
=
Example:
cos
sin
.
θ
θ
1
sec
cos
θ
cos
θ
θ
+
2
2
1
1
cos
sin
1
θ
θ
+
=
+
=
=
2
2
Example:
csc
sec
.
θ
θ
θ
θ
θ
θ
2
2
2
2
2
2
sin
cos
sin
cos
sin
cos
Problems:
Write each expression in terms of the sine and cosine functions and then simplify the expression.
θ
θ
−
2
sec
θ
θ
csc
1
114. sec
+ tan
116.
115.
θ
θ
+
θ
tan
cot
cot
Here are some suggestions for verifying trigonometric identities successfully.
1. Become very familiar with the reciprocal, quotient and Pythagorean identities in all their forms.
2. Begin with the more complicated side of the identity and, using the basic identities, try to
reduce it to the simpler side.
3. If necessary, convert all functions to the sine and cosine and then proceed to simplify each side.
θ
θ
θ
2
4
4
Example: Verify the identity 2 sec
- 1 = sec
- tan
.
Solution: Begin with the more complicated right-hand side and factor it.
(
) (
)
θ
θ
θ
θ
θ
θ
−
=
+
−
4
4
2
2
2
2
sec
tan
sec
tan
sec
tan
(
)
θ
θ
θ
θ
=
+
⋅
=
+
2
2
2
2
sec
tan
1
sec
tan
(
)
θ
θ
θ
=
+
−
=
−
2
2
2
sec
sec
1
2
sec
1
θ
θ
θ
θ
Example: Verify the identity tan
+ cot
= csc
sec
.
Solution: Since no terms are squared, the Pythagorean identities are not helpful. Begin by writing
the left side in terms of the sine and cosine.
θ
θ
θ
θ
+
2
2
sin
cos
sin
cos
1
θ
θ
+
=
+
=
=
tan
cot
θ
θ
θ
θ
θ
θ
cos
sin
cos
sin
cos
sin
1
1
θ
θ
=
⋅
=
⋅
csc
sec
θ
θ
sin
cos
Problems:
Verify each of the following identities.
θ
θ
θ
θ
θ
θ
2
2
2
4
2
4
117. cos
- sin
= 2 cos
- 1
118. cot
+ 2 cot
+ 1 = csc
- 26 -
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