# Trigonometry Formulas

November 15, 2017 | Author: AaRichard Manalo | Category: Trigonometric Functions, Sine, Sphere, Triangle, Trigonometry

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CE Review - Trigonometry...

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PRINCIPLES IN FUNCTION OF RIGHT TRIANGLE

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

Opposite, a

Reciprocal and Quotient

π½ Adjacent, b

FUNCTION OF ANY ANGLE π½

csc ΞΈ =

1 , provided sin ΞΈ β  0 sin ΞΈ

sec ΞΈ =

1 , provided cos ΞΈ β  0 cos ΞΈ

cot ΞΈ =

1 , provided tan ΞΈ β  0 tan ΞΈ

tan ΞΈ =

sin ΞΈ , provided cos ΞΈ β  0 cos ΞΈ

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

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

SIGN OF FUNCTION VALUES

cos x β y = cosxcosy β sinxsiny tan x + y =

tanx + tany 1 β tanxtany

tan x β y =

tanx β tany 1 + tanxtany

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

2tanx 1 β 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 β cos x 2

x cos = 2

1 + cos x 2

x

tan 2 =

1 βcos x sin x

sin x

= 1+cos x =

1βcos 2x

sin π₯ + sin π¦ = 2 sin

π₯+π¦ π₯βπ¦ cos 2 2

sin π₯ β sin π¦ = 2 cos

π₯+π¦ π₯βπ¦ sin 2 2

cos π₯ + cos π¦ = 2 cos

π₯+π¦ π₯βπ¦ cos 2 2

cos π₯ β cos π¦ = β2 sin

1+cos 2x

tan π₯ + tan π¦ =

sin π₯ + π¦ cos π₯ cos π¦

tan π₯ β tan π¦ =

sin π₯ β π¦ cos π₯ cos π¦

Power of Functions

π ππ2 π₯ =

1 β πππ 2π₯ 2

πππ  2 π₯ =

1 + πππ 2π₯ 2

2

π₯+π¦ π₯βπ¦ sin 2 2

SOLUTION OF TRIANGLES

C

1β πππ 2π₯

π‘ππ π₯ = 1+ πππ 2π₯

b

a Product of Functions

1 π πππ₯πππ π¦ = sin π₯ + π¦ + sin π₯ β π¦ 2

B

c

Sine Law a b c = = sinA sinB sinC

1 π πππ₯π πππ¦ = cos π₯ β π¦ β cos π₯ + π¦ 2 πππ π₯πππ π¦ =

1 cos π₯ + π¦ + cos π₯ β π¦ 2

A

Cosine Law a2 = b2 + c 2 β 2bccosA b2 = a2 + c 2 β 2accosB c 2 = a2 + b2 β 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 Napierβs Rules

Sum of interior angles of spherical triangle

0

angle. Then apply

π΄ = 90 β A π΅ = 90 β B π = 90 βc

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

NAPIERβ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 Napierβs circle, the sine of any middle part is equal to product of the cosines of its opposite parts. SIN-TAAD Rule in the Napierβ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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