Summary MTH101 trigonometry formula

Trigonometry Formula Sheet:
Arc Length:
𝑠 = 𝜃𝑟 𝜃 → 𝑟𝑎𝑑𝑖𝑎𝑛𝑠

Area of a Sector:
1 2
𝐴= 𝜃𝑟 𝜃 → 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
2

𝜃
𝐴= ( ) 𝜋𝑟 2 𝜃 → 𝑑𝑒𝑔𝑟𝑒𝑒𝑠
360°
Six Trig Functions: (SOH CAH TOA)

𝑜𝑝𝑝 ℎ𝑦𝑝
sin 𝜃 = csc 𝜃 =
ℎ𝑦𝑝 𝑜𝑝𝑝

𝑎𝑑𝑗 ℎ𝑦𝑝
cos 𝜃 = sec 𝜃 =
ℎ𝑦𝑝 𝑎𝑑𝑗

𝑜𝑝𝑝 𝑎𝑑𝑗
tan 𝜃 = cot 𝜃 =
𝑎𝑑𝑗 𝑜𝑝𝑝
Graphing Trig Functions:

𝑦 = 𝐴𝑠𝑖𝑛(𝐵𝑥 + 𝑐 ) + 𝐷

max − 𝑚𝑖𝑛 2𝜋
Amplitude: |𝐴| = Period: 𝑃 =
2 𝐵

max + 𝑚𝑖𝑛 −𝐶
Vertical Shift: 𝐷 = Phase Shift: 𝑥 =
2 𝐵

Law of Sines:
sin 𝐴 sin 𝐵 sin 𝐶
= =
𝑎 𝑏 𝑐

Law of Cosines:

𝑐 2 = 𝑎2 + 𝑏2 + 2𝑎𝑏 cos 𝐶

Law of Tangents:

𝑎 − 𝑏 tan[1⁄2 (𝐴 − 𝐵)]
=
𝑎 + 𝑏 tan[1⁄2 (𝐴 + 𝐵)]


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, Reciprocal Identities: Quotient Identities: Pythagorean Identities:

1 sin 𝜃 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1
cot 𝜃 = tan 𝜃 =
tan 𝜃 cos 𝜃
1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃
1 cos 𝜃
csc 𝜃 = cot 𝜃 =
sin 𝜃 sin 𝜃 1 + 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑠𝑐 2 𝜃
1
sec 𝜃 =
cos 𝜃

Even-Odd Identities: Co-function Identities: Power Reducing Formulas:

sin(−𝜃) = − sin 𝜃 cos(90° − 𝜃) = sin 𝜃 1 − cos 2𝜃
𝑠𝑖𝑛2 𝜃 =
cos(−𝜃) = cos 𝜃 sin(90° − 𝜃) = cos 𝜃 2
tan(−𝜃) = − tan 𝜃 tan(90° − 𝜃) = cot 𝜃
csc(−𝜃) = − csc 𝜃 cot(90° − 𝜃) = tan 𝜃 1 + cos 2𝜃
𝑐𝑜𝑠 2 𝜃 =
sec(−𝜃) = sec 𝜃 sec(90° − 𝜃) = csc 𝜃 2
cot(−𝜃) = −𝑐𝑜𝑡𝜃 csc(90° − 𝜃) = sec 𝜃
1 − cos 2𝜃
𝑡𝑎𝑛2 𝜃 =
1 + cos 2𝜃
Double Angle Formulas: Half-Angle Formulas: Triple Angle Formulas:

sin 2𝜃 = 2 sin 𝜃 cos 𝜃
𝜃 1 − cos 𝜃
sin 3𝜃 = 3 sin 𝜃 − 4𝑠𝑖𝑛3 𝜃
𝑠𝑖𝑛 ( ) = ±√
2 tan 𝜃 2 2
sin 2𝜃 =
1+𝑡𝑎𝑛2 𝜃
__________________________ ____________________________ cos 3𝜃 = 4𝑐𝑜𝑠 3 𝜃 − 3 cos 𝜃
cos 2𝜃 = 𝑐𝑜𝑠 2 𝜃 − 𝑠𝑖𝑛2 𝜃
𝜃 1 + cos 𝜃
cos ( ) = ±√
2 2 2 3 tan 𝜃 − 𝑡𝑎𝑛3 𝜃
cos 2𝜃 = 2𝑐𝑜𝑠 − 1 tan 3𝜃 =
1 − 3𝑡𝑎𝑛2 𝜃
cos 2𝜃 = 1 − 2𝑠𝑖𝑛2 𝜃
𝜃 1 − cos 𝜃 sin 𝜃
tan ( ) = =
1−𝑡𝑎𝑛2 𝜃 2 2 1 + cos 𝜃
cos 2𝜃 =
1+𝑡𝑎𝑛2 𝜃
__________________________
2 tan 𝜃
tan 2𝜃 = 𝜃 1 − cos 𝜃
1−𝑡𝑎𝑛2 𝜃 tan ( ) = ±√
2 1 + cos 𝜃



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