Fundamental Identities: These identities are consequences of the definitions and the Pythagorean
Theorem. You should already understand how to get these identities before we get to chapter 5
(Identities).
Definitions: For an acute angle Θ in a right triangle we define the following trig ratios:
(Θc is the complement of Θ)
Sine Θ = sin Θ = opposite/hypotenuse
Cosine Θ = cos Θ = adjacent/hypotenuse
Tangent Θ = tan Θ = opposite/adjacent
Cotangent Θ =cot Θ = adjacent/hypotenuse
Secant Θ = sec Θ = hypotenuse/adjacent
Cosecant Θ = csc Θ = hypotenuse/opposite.
We can immediately conclude these identities:
[Remember that Θc = 90o- Θ = 90o- Θ = π/2 – Θ]
Cofunction Identities
cos Θ = sin Θc and sin Θ = cos Θc
cot Θ = tan Θc and tan Θ = cot Θc
csc Θ = sec Θc and sec Θ = csc Θc
Sin and Cos Ratios
tan Θ = sin Θ/cos Θ
cot Θ = cos Θ/sin Θ
sec Θ = 1/cos Θ
csc Θ = 1/sin Θ
Ratio Identities
csc Θ = 1/sin Θ and sin Θ = 1/csc Θ
sec Θ = 1/cos Θ and cos Θ = 1/sec Θ
tan Θ = 1/ cot Θ and cot Θ = 1/ tan Θ
We extend our definition of trig functions to apply to all real numbers by use of the unit circle as follows:
From this we can deduce the symmetry identities:
Negative angle symmetry Identities
sin(–Θ) = – sin Θ
cos(–Θ) =+ cos Θ
tan(–Θ) = – tan Θ
cot(–Θ) = – cot Θ
sec(–Θ) = +sec Θ
csc(–Θ) = – csc Θ
Supplementary identities
sin(180o–Θ) = +sin Θ
cos(180o–Θ) = – cos Θ
tan(180o–Θ) = – tan Θ
cot(180o–Θ) = – cot Θ
sec(180o–Θ) = – sec Θ
csc(180o–Θ) = + csc Θ
Plus Half Circle Identities
sin(Θ+180o) = –sin Θ
cos(Θ+180o) = –cos Θ
tan(Θ+180o) = +tan Θ
cot(Θ+180o) = +cot Θ
sec(Θ+180o) = –sec Θ
csc(Θ+180o) = –csc Θ
From the Pythagorean Theorem:
(opposite)2+( adjacent)2=(hypotenuse)2
We deduce the following about sine and cosine
(sin Θ)2 + (cos Θ)2 = 1
We usually write this as sin2Θ + cos2Θ = 1.
After some algebra we get the Pythagorean identities:
Pythagorean Identities
sin2Θ+cos2Θ = 1
tan2Θ + 1 = sec2Θ
1 + cot2Θ = csc2Θ
This concludes most of the fundamental identities.