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