Section 7.2: Right Triangle Trigonometry
• Def: The trigonometric functions of acute angles are the six ratios which can
be obtained from a right triangle. The six trigonometric functions are defined
as follows:
opposite
hypotenuse
adjacent
cosine of θ = cos θ = ac = hypotenuse
opposite
tangent of θ = tan θ = ab = adjacent
cosecant of θ = csc θ = cb = hypotenuse
opposite
hypotenuse
c
secant of θ = sec θ = a = adjacent
cotangent of θ = cot θ = ab = adjacent
opposite
1. sine of θ = sin θ =
2.
3.
4.
5.
6.
b
c
=
• ex. Find the value of the six trigonometric functions of the angle θ in the
figure.
• Among the six trigonometric functions, there are some relationships between
some of them.
– Reciprocal Identities:
csc θ =
1
sin θ
sec θ =
1
cos θ
cot θ =
1
tan θ
sin θ =
1
csc θ
cos θ =
1
sec θ
tan θ =
1
cot θ
1
– Quotient Identities:
tan θ =
sin θ
cos θ
cot θ =
cos θ
sin θ
• ex. Use the definition or identities to find the exact value of each of the
remaining five trigonometric functions of the acute angle θ.
√
(a) cos θ =
2
4
(b) cot θ = 3
• Pythagorean Identities:
sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
cot2 θ + 1 = csc2 θ
• Note: The second and third of the Pythagorean identities are obtained from
the first identity by dividing each term by either cos2 θ or by sin2 θ, respectively, and using the reciprocal or quotient identities to simplify.
• Collectively, the reciprocal identities, the quotient identities, and the Pythagorean
identities are called the Fundamental identities.
• Def: Two acute angles of a right triangle are called complementary if their
sum is 90◦ . In the diagram below, the angles α and β are complementary
angles.
2
• Note: For the complementary angles α and β, the following relationships
between the trigonometric functions exist:
sin α =
b
c
= cos β
cos α =
a
c
= sin β
tan α =
b
a
= cot β
csc α =
c
b
= sec β
sec α =
c
a
= csc β
cot α =
a
b
= tan β
• Def: Trigonometric functions which are related by having the same value
at complementary angles are called cofunctions. Thus, sine and cosine are
cofunctions, cosecant and secant are cofunctions, and tangent and cotangent
are cofunctions.
• Complementary Angle Theorem: Cofunctions of complementary angles are
equal.
• The Complementary Angle Theorem just says in words what the relationships
between the trigonometric functions of complementary angles above say in
equations.
• Another way of stating the Complementary Angle Theorem is given by the
following relationships (each relation is stated for θ given in degrees or in
radians):
sin θ = cos (90◦ − θ)
sin θ = cos π2 − θ
cos θ = sin (90◦ − θ)
cos θ = sin
π
2
−θ
tan θ = cot (90◦ − θ)
tan θ = cot
π
2
−θ
csc θ = sec (90◦ − θ)
csc θ = sec
π
2
−θ
sec θ = csc (90◦ − θ)
sec θ = csc
π
2
−θ
cot θ = tan (90◦ − θ)
cot θ = tan
π
2
−θ
• Use Fundamental Identities and/or the Complementary Angle Theorem to
find the exact value of each expression.
(a) csc2 40◦ − cot2 40◦
3
(b)
sin 38◦
cos 52◦
(c) sin 40◦ · csc 50◦ · cot 40◦
• ex. Given cos 60◦ =
of
√
3
,
2
use trigonometric identities to find the exact value
(a) sin 30◦
(b) sin2 60◦
(c) csc π6
(d) sec π3
4