6.3 A
Trigonometric Identities
Objectives: Use algebra to simplify trigonometric expressions & establish identities
Title: Mar 10­9:51 AM (1 of 11)
Trig Functions
Tan( θ) = Sin( θ)
Cos( θ)
Cot(θ) = Cos(θ) = 1
Sin(θ) Tan(θ)
Csc(θ) = 1
Sin(θ)
Sec(θ) = 1
Cos(θ)
The COsecant goes with Sine. The Secant goes with COsine.
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Pythagorean Theorem and Trig!!!
x2 + y2 = 1
y2 + x2 = 1
sin2 θ + cos2 θ = 1
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Pythagorean Theorem and Trig!!!
Divide by cos2 θ
sin2 θ
cos2 θ
+
cos2 θ
cos2 θ
Divide by sin2 θ
= cos2 θ
tan2 θ + 1 = sec2 θ
Title: Mar 10­9:51 AM (4 of 11)
1
sin2 θ
sin2 θ
+
cos2 θ
sin2 θ
1
= sin2 θ
1 + cot2θ = csc2 θ
Reset
90
Notice for sine of an angle the sine of the negative angle is just the negative value.
120
60
45
135
150
30
sin(5π
/6) = √
3/2
cos(5π
/6) = √
3/2
cos(5π/6) = √3/2
5π
/6
5π
/6
π/4 cos(π/4) = √
sin(π/4) = √2/2
2/2360
5π/6
cos(π/4) = √
sin(­π/4) = ­√2/2
2/2
­5π
//66 ­π/4
­5π
­5π
/6
sin(­5π
/6) = ­√
3/2
cos(5π
/6) = √
3/2
Example 1
Example 2
180
Notice for cosine if you take the cosine of a negative angle it is the same as the cosine of the positive angle.
210
cos(5π/6) = √3/2
225
Example 3
Example 4
Example 4
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330
315
300
240
270
Even­Odd Identitites
­π/3 = 5π/3
Sin(π/3) = √3/2
Sin(­π/3) = -√3
/2
π/3 or 60o
225o
­7π/6
3=
-π/
­(­7π/3) = 7π/3
­2
­225
= 135
25
o
/3) = -1/2
­7Sin(­7π
π/6
Sin(225) = -√2/2
Sin(­(­7π/3)) = 1/2
Sin(­225) = √2/2
Pick an angle and we will see what
the negative angle does to the sine and cosine.
Cos(­π/3) = 1/2
22
Title: Mar 10­9:51 AM (6 of 11)
π/
3
5o
6)
7π/
­
(
­
Reset
3
5π/
Cos(π/3) = 1/2
Cos(7π/3) = -√3
/2
Cos(225) = -√2/2
Cos(­(­7π/3)) = -√3
/2
Cos(­225) = -√2/2
Even­Odd Identitites
Sin(­θ) = ­Sin(θ)
Csc(­θ) = 1 = 1 = ­Csc( θ)
Sin(­ θ) ­Sin( θ)
Csc(­θ) = ­Csc(θ)
Sec(­θ) = 1 = 1 = Sec( θ) Cos(­θ) Cos(θ) Cos(­θ) = Cos(θ)
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Sec(­θ) = Sec( θ)
Even­Odd Identitites
Tan(­θ) = Sin(­ θ) = ­Sin(θ) = ­Tan(θ)
Cos(­θ) Cos(θ)
Tan(­θ) = ­Tan(θ)
Cot(­θ) = Cos(­θ) = Cos(θ) = ­Cot( θ) Sin(­ θ) ­Sin(θ) Cot(­θ) = ­Cot( θ) Title: Mar 10­9:51 AM (8 of 11)
Example 1
Multiply:
tan(θ) cos(θ) + 1
Title: Mar 10­9:51 AM (9 of 11)
cos(θ) ­ 1
cos(θ) ­ 1
Distribute top
Foil Bottom
tan(θ)cos(θ) ­ tan(θ)
cos2(θ) ­ cos(θ) + cos (θ) ­1 Rewrite top as parts
Simplify Bottom
sin(θ)
/cos(θ)
cos(θ) ­ sin(θ)
/cos(θ)
cos2(θ) ­1
Multiply / Simplify Top
Identity!! on the Bottom
sin(θ) ­ sin(θ)
/cos(θ)
sin2(θ)
Factor out sine
sin(θ)(1­ 1/cos(θ)
)
sin2(θ)
Simplify
1­ 1/cos(θ)
sin(θ)
Rewrite Trig funtion
1 ­ sec(θ)
sin(θ)
Done!! Example 2
Rewrite over a common denominator:
cot(θ) + tan(θ)
cos(θ)
sin(θ)
+
sin(θ)
cos(θ)
Find common denominator
cos(θ) cos(θ) sin(θ) sin(θ)
+
sin(θ) cos(θ) cos(θ) sin(θ)
Multipy Top and Bottom
+
cos2(θ) sin2(θ)
sin(θ)cos(θ) sin(θ)cos(θ)
Add fractions
cos2(θ) + sin2(θ)
sin(θ)cos(θ)
Identity!!!
1
sin(θ)cos(θ)
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Rewrite as parts
Done!
Homework:
page 471
(9 - 16)
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