Trigonometry Formulas

PRINCIPLES IN FUNCTION OF RIGHT TRIANGLE

M A T H E M A T I C S TRIGONOMETRIC IDENTITIES

Opposite, a

Reciprocal and Quotient

𝜽 Adjacent, b

FUNCTION OF ANY ANGLE 𝜽

csc θ =

1 , provided sin θ ≠ 0 sin θ

sec θ =

1 , provided cos θ ≠ 0 cos θ

cot θ =

1 , provided tan θ ≠ 0 tan θ

tan θ =

sin θ , provided cos θ ≠ 0 cos θ

The Pythagorean Identities sin2 θ + cos 2 θ = 1 tan2 θ + 1 = sec 2 θ cot 2 θ + 1 = csc 2 θ

Sum & Difference of Two Angles sin x + y = sinxcosy + cosxsiny sin x − y = sinxcosy − cosxsiny cos x + y = cosxcosy − sinxsiny

SIGN OF FUNCTION VALUES

cos x − y = cosxcosy − sinxsiny tan x + y =

tanx + tany 1 − tanxtany

tan x − y =

tanx − tany 1 + tanxtany

Sum & Difference of Two Angles sin 2x = 2sinxcox cos2x = cos 2 x − sin2 x = 1 − 2sin2 x = 2 cos 2 x -1 tan2x =

2tanx 1 − tan2 x

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PRINCIPLES IN

M A T H E M A T I C S

TRIGONOMETRIC IDENTITIES

TRIGONOMETRIC IDENTITIES

Half-Angle Identity

Sum and Difference of Function

x sin = 2

1 − cos x 2

x cos = 2

1 + cos x 2

x

tan 2 =

1 −cos x sin x

sin x

= 1+cos x =

1−cos 2x

sin 𝑥 + sin 𝑦 = 2 sin

𝑥+𝑦 𝑥−𝑦 cos 2 2

sin 𝑥 − sin 𝑦 = 2 cos

𝑥+𝑦 𝑥−𝑦 sin 2 2

cos 𝑥 + cos 𝑦 = 2 cos

𝑥+𝑦 𝑥−𝑦 cos 2 2

cos 𝑥 − cos 𝑦 = −2 sin

1+cos 2x

tan 𝑥 + tan 𝑦 =

sin 𝑥 + 𝑦 cos 𝑥 cos 𝑦

tan 𝑥 − tan 𝑦 =

sin 𝑥 − 𝑦 cos 𝑥 cos 𝑦

Power of Functions

𝑠𝑖𝑛2 𝑥 =

1 − 𝑐𝑜𝑠2𝑥 2

𝑐𝑜𝑠 2 𝑥 =

1 + 𝑐𝑜𝑠2𝑥 2

2

𝑥+𝑦 𝑥−𝑦 sin 2 2

SOLUTION OF TRIANGLES

C

1− 𝑐𝑜𝑠2𝑥

𝑡𝑎𝑛 𝑥 = 1+ 𝑐𝑜𝑠2𝑥

b

a Product of Functions

1 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑦 = sin 𝑥 + 𝑦 + sin 𝑥 − 𝑦 2

B

c

Sine Law a b c = = sinA sinB sinC

1 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦 = cos 𝑥 − 𝑦 − cos 𝑥 + 𝑦 2 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 =

1 cos 𝑥 + 𝑦 + cos 𝑥 − 𝑦 2

A

Cosine Law a2 = b2 + c 2 − 2bccosA b2 = a2 + c 2 − 2accosB c 2 = a2 + b2 − 2abcosC

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PRINCIPLES IN

M A T H E M A T I C S

SPHERICAL TRIGONOMETRY

Spherical Excess The area of a spherical triangle on the surface of the sphere of radius R is given by the

A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. Note that for spherical triangles, sides a, b, and c are usually in angular units. And like plane triangles, angles A, B, and C are also in angular units

RIGHT SPHERICAL TRIANGLE

To solve a right triangle, draw a circle with 5 parts. The 5 parts corresponds to the 3 sides and 2 angles of the triangle (excluding the 90 Napier’s Rules

Sum of interior angles of spherical triangle

0

angle. Then apply

𝐴 = 90 − A 𝐵 = 90 − B 𝑐 = 90 −c

The sum of the interior angles of a spherical triangle is greater than 180° and less than 540

NAPIER’S RULE Area of spherical triangle The area of a spherical triangle on the surface of the sphere of radius R is given by the

SIN-COOP Rule in the Napier’s circle, the sine of any middle part is equal to product of the cosines of its opposite parts. SIN-TAAD Rule in the Napier’s circle, the sine of any middle part is equal to the product of the tangents of its adjacent parts.

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PRINCIPLES IN

M A T H E M A T I C S

OBLIQUE TRIANGLE

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