Tutorial Services – Mission del Paso Campus

1 Trigonometric Identities & Formulas Tutorial Services Mission del Paso Campus Reciprocal Identities Ratio or Quotient Identities 1 1 sin x cos x sin x csc x tan x cot x . csc x sin x cos x sin x 1 1. cos x sec x sinx = cosx tanx cosx = sinx cotx sec x cos x 1 1. tan x cot x . cot x tan x Pythagorean Identities Pythagorean Identities in Radical Form sin x cos x 1. 2 2. sin x 1 cos2 x 1 tan 2 x sec2 x 1 cot 2 x csc2 x tan x sec 2 x 1. Note: there are only three, basic Pythagorean identities, the other forms cos x 1 sin 2 x are the same three identities, just arranged in a different order.

2 Confunction Identities Odd-Even Identities Also called negative angle identities . sin x cos x cos x sin x Sin (-x) = -sin x Csc (-x) = -csc x 2 2 . Cos (-x) = cos x Sec (-x) = sec x . tan x cot x cot x tan x Tan (-x) = -tan x Cot (-x) = -cot x 2 2 . c sec x csc x csc x sec x Phase Shift =. 2 2 b 2 . Period =. b Sum and Difference Formulas/Identities How to Find Reference Angles sin(u v ) sin u cos v cos u sin v Step 1: Determine which quadrant the angle is in sin(u v ) sin u cos v cos u sin v Step 2: Use the appropriate formula Quad I = is the angle itself cos(u v ) cos u cos v sin u sin v Quad II = 180 or.

3 Cos(u v ) cos u cos v sin u sin v Quad III = 180 or - . Quad IV = 360 or 2 - . tan u tan v tan(u v ) . 1 tan u tan v tan u tan v tan(u v ) . 1 tan u tan v Saved C: Trigonometry Formulas {Web Page} microsoftword & PDF. Website: 1. Reciprocal Identities Ratio or Quotient Identities 1 1 sin x cos x sin x csc x tan x cot x . csc x sin x cos x sin x 1 1. cos x sec x sinx = cosx tanx cosx = sinx cotx sec x cos x 1 1. tan x cot x . cot x tan x Pythagorean Identities Pythagorean Identities in Radical Form sin x cos x 1. 2 2. sin x 1 cos2 x 1 tan 2 x sec2 x 1 cot 2 x csc2 x tan x sec 2 x 1. Note: there are only three, basic Pythagorean identities, the other forms are the same three identities, just arranged in a different order.

4 Confunction Identities Odd-Even Identities Also called negative angle identities . sin x cos x cos x sin x Sin (-x) = -sin x Csc (-x) = -csc x 2 2 . Cos (-x) = cos x Sec (-x) = sec x .. tan x cot x cot x tan x Tan (-x) = -tan x Cot (-x) = -cot x 2 2 .. sec x csc x csc x sec x 2 2 . Sum and Difference Formulas - Identities sin(u v ) sin u cos v cos u sin v cos(u v ) cos u cos v sin u sin v sin(u v ) sin u cos v cos u sin v cos(u v ) cos u cos v sin u sin v tan u tan v tan u tan v tan(u v ) tan(u v ) . 1 tan u tan v 1 tan u tan v Saved C: Trigonometry Formulas {Web Page} microsoftword & PDF.

5 Website: 2. The Unit Circle 90 . 3 3. Tan = - 3 cot tan = undefined & cot= 0 tan = 3 cot =. 3 3. 120 60 . Tan = 1. Tan =- - 1 cot = 1. Cot = -1. 135 45 . 150 30 ..785. 3 3. Tan = cot = - 3 .523 tan = cot = 3. 3 3. Tan= 0 Tan=0 & cot=undef Cot=undef 180 360 . 2( )= 3 3. Tan cot = 3 tan = cot = - 3. 3 3. 330 . 210 . Tan = -1. Tan = 1 Cot = -1. Cot = 1. 225 315 . 240 270 300 . 3 3. Tan = 3 cot = tan=undefined tan = - 3 cot = . 3 3. Cot = 0. Definition of Trigonometric Functions concerning the Unit Circle opp y hyp r sin = csc = . hyp r opp y adj x hyp r cos = sec = . hyp r adj x opp y adj x tan = cot =.

6 Adj x opp y Saved C: Trigonometry Formulas {Web Page} microsoftword & PDF. Website: 3. Right Triangle Definitions of Trigonometric Functions Note: sin & cos are complementary angles, so are tan & cot and sec & cos, and the sum of complementary angles is 90 degrees. C. opp y hyp r sin = csc = . hyp r opp y r y Hypotenuse opposite adj x hyp r cos = sec = . hyp r adj x A x B. adjacent opp y adj x tan = cot = . adj x opp y Adjacent = is the side adjacent to the angle in consideration. So if we are considering Angle A, then the adjacent side is CB. Trigonometric Values of Special Angles Degrees 0 30 45 60 90 180 270.

7 Radians 0 6 4 3 2 3 . 2. 1 2 3. sin 0 2 2 2 1 0 -1. 3 2 1. cos 1 2 2 2 0 -1 0. 3. tan 0 1 3 undefined 0 undefined 3. rad To Convert Degrees to Radians, Multiply by 180deg 180deg To Convert Radians to Degrees, Multiply by rad Vocabulary Cotangent Angles - are two angles with the same terminal side Reference Angle - is an acute angle formed by terminal side of angle( ) with x-axis Saved C: Trigonometry Formulas {Web Page} microsoftword & PDF. Website: 4. double Angle Identities half Angle Identities Power Reducing Formulas A 1 cos A 1 cos 2u sin 2 A 2 sin A cos A sin sin 2 u . 2 2 2.

8 A 1 cos A 1 cos 2u cos 2 A cos2 A sin 2 A cos cos2 u . 2 2 2. A 1 cos A 1 cos 2u cos 2 A 2 cos2 A 1 tan tan 2 u . 2 sin A 1 cos 2u cos 2 A 1 2 sin 2 A. 2 tan A A sin A. tan 2 A tan . 1 tan 2 A 2 1 cos A. Product-to-Sum Formulas Sum-to-Product Formulas 1 x y x y . sin u sin v cos(u v ) cos(u v ) sin x sin y 2 sin cos . 2 2 2 . 1 x y x y . cos u cos v . 2. cos(u v) cos(u v) sin x sin y 2 cos sin . 2 2 .. 1 x y x y . sin u cos v . 2. sin(u v) sin(u v) cos x cos y 2 cos . 2 . cos . 2 .. 1 x y x y . cos u sin v . 2. sin(u v) sin(u v) cos x cos y 2 sin sin . 2 2 .. Law of Sines Law of Cosines Solving Oblique Triangles using sine: AAS, ASA, SSA, SSS, SAS Cosine: SAS, SSS.

9 Standard Form Alternative Form a b c sin A sin B sin C b2 c2 a 2. or a b c 2bc cos A. 2 2 2. cos A . sin A sin B sin C a b c 2bc a c2 b2. 2. b 2 a 2 c 2 2ac cos B cos B . 2ac a b 2 c2. 2. c 2 b 2 a 2 2ab cos C cosC . 2ab Finding the Area of non-90degree Triangles Area of an Oblique Triangle Heron's formula 1 1 1 a b c . area bc sin A ab sin C ac sin B Step 1: Find s s . 2 2 2 2. Step 2: Use the formula area s( s a )( s b)( s c). Saved C: Trigonometry Formulas {Web Page} microsoftword & PDF. Website: 5.