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Chapter 13 Periodic Functions and Trigonometry

Convert Between Radians and Degrees

Use the proportion d ° 180 ° = r  radians π  radians fraction d degrees , over 180 degrees end fraction . equals . fraction r , radians , over pi , radians end fraction to convert between radians and degrees.

To convert degrees to radians, multiply by π  radians 180 ° . fraction pi , radians , over 180 degrees end fraction . .

To convert radians to degrees, multiply by 180 ∘ π  radians . fraction 180 composition , over pi , radians end fraction . .

Length of an Intercepted Arc

For a circle of radius r and a central angle of measure θ theta (in radians), the length s of the intercepted arc is s = r θ . s equals r theta .

Sine and Cosine Functions

	Sine	Cosine
Parents	y = sin x y equals sine x	y = cos x y equals cosine x
Reflection across x-axis	y = − sin x y equals negative sine x	y = − cos x y equals negative cosine x
Amplitude |a|	y = a sin x y equals eh sine x	y = a cos x y equals eh cosine x
Period 2 π b , b > 0 fraction 2 pi , over b end fraction , comma . b greater than 0	y = sin b x y equals sine b x	y = cos b x y equals cosine b x
Translation
horizontal by h vertical by k	y = sin ( x − h ) + k y equals sine open x minus h close plus k	y = cos ( x − h ) + s k y equals cosine open x minus h close plus s k

Tangent Function

Parent	y = tan x y equals tangent x
Reflection across x-axis	y = − tan x y equals negative tangent x
Period π b pi over b	y = tan b x y equals tangent b x
Translation
horizontal by h vertical by k	y = tan ( x − h ) + k y equals tangent open x minus h close plus k
Asymptotes (tan bx)	x = n π 2 b , x equals n , fraction pi , over 2 b end fraction , comma n odd

Chapter 14 Trigonometric Identities and Equations

Basic Identities

Reciprocal Identities:

csc θ = 1 sin θ sec θ = 1 cos θ tan θ = 1 cot θ sin θ = 1 csc θ cos θ = 1 sin θ cot θ = 1 tan θ table with 2 rows and 3 columns , row1 column 1 , co-secant theta equals . fraction 1 , over sine theta end fraction , column 2 secant theta equals . fraction 1 , over cosine theta end fraction , column 3 tangent theta equals . fraction 1 , over co-tangent theta end fraction , row2 column 1 , sine theta equals . fraction 1 , over co-secant theta end fraction , column 2 cosine theta equals . fraction 1 , over sine theta end fraction , column 3 co-tangent theta equals . fraction 1 , over tangent theta end fraction , end table

Tangent Identity: tan θ = sin θ cos θ tangent theta equals . fraction sine theta , over cosine theta end fraction

Cotangent Identity: cot θ = cos θ sin θ co-tangent theta equals . fraction cosine theta , over sine theta end fraction

Pythagorean Identities

cos 2 θ + sin 2 θ = 1 cosine squared , theta plus , sine squared , theta equals 1

1 + tan 2 θ = sec 2 θ 1 plus , tangent squared , theta equals , secant squared , theta

cot 2 θ + 1 = csc 2 θ co-tangent squared , theta plus 1 equals , co-secant squared , theta

Area of a Triangle

In Δ A B C cap delta eh b c with a, b, and c the lengths of the sides opposite ∠ A , ∠ B , angle eh comma angle b comma and ∠ C , angle c comma respectively,
Δ A B C = 1 2 b c   sin A = 1 2 a c   sin   B = 1 2 a b   sin C cap delta eh b c equals , 1 half , b c sine eh equals , 1 half , eh c sine b equals , 1 half , eh b sine c

Law of Sines

In Δ A B C cap delta eh b c with a, b, and c the lengths of the sides opposite ∠ A , ∠ B , angle eh comma angle b comma and ∠ C , angle c comma respectively,
sin A a = sin B b = sin C c . fraction sine eh , over eh end fraction . equals . fraction sine b , over b end fraction . equals . fraction sine c , over c end fraction . .

Law of Cosines

In Δ A B C cap delta eh b c with a, b, and c the lengths of the sides opposite ∠ A , ∠ B , angle eh comma angle b comma and ∠ C , angle c comma respectively,

a 2 = b 2 + c 2 − 2 b c · cos A eh squared , equals , b squared , plus , c squared , minus 2 b c middle dot cosine eh

b 2 = a 2 + c 2 − 2 a c · cos B b squared , equals , eh squared , plus , c squared , minus 2 eh c middle dot cosine b

c 2 = a 2 + b 2 − 2 a b · cos C c squared , equals , eh squared , plus , b squared , minus 2 eh b middle dot cosine c

Negative Angle Identities

sin ( − θ ) = − sin θ sine open negative theta close equals negative sine theta

cos ( − θ ) = cos θ cosine open negative theta close equals cosine theta

tan ( − θ ) = − tan θ tangent open negative theta close equals negative tangent theta

Cofunction Angle Identities

sin ( π 2 − θ ) = cos   θ   cos ( π 2 − θ ) = sin   θ   tan ( π 2 − θ ) = cot   θ sine . open . pi over 2 , minus theta . close . equals cosine theta cosine . open . pi over 2 , minus theta . close . equals sine theta tangent . open . pi over 2 , minus theta . close . equals co-tangent theta

Angle Difference Identities

sin ( A − B ) = sin A cos B − cos A sin B sine open eh minus b close equals sine eh cosine b minus cosine eh sine b

cos ( A − B ) = cos A cos B + sin A sin B cosine open eh minus b close equals cosine eh cosine b plus sine eh sine b

tan ( A − B ) = tan A − tan B 1 + tan A tan B tangent . open , eh minus b , close . equals . fraction tangent eh minus tangent b , over 1 plus tangent eh tangent b end fraction

Angle Sum Identities

sin ( A + B ) = sin A cos B + cos A sin B sine open eh plus b close equals sine eh cosine b plus cosine eh sine b

cos ( A + B ) = cos A cos B − sin A sin B cosine open eh plus b close equals cosine eh cosine b minus sine eh sine b

tan ( A + B ) = tan A − tan B 1 − tan A tan B tangent . open , eh plus b , close . equals . fraction tangent eh minus tangent b , over 1 minus tangent eh tangent b end fraction

Double-Angle Identities

cos 2 θ = cos 2 θ − sin 2 θ cosine 2 theta equals , cosine squared , theta negative , sine squared , theta

cos 2 θ = 2 cos 2 θ − 1 cosine 2 theta equals 2 , cosine squared , theta negative 1

cos 2 θ = 1 − 2 sin 2 θ cosine 2 theta equals 1 minus 2 , sine squared , theta

sin 2 θ = 2 sin θ cos θ sine 2 theta equals 2 sine theta cosine theta

tan 2 θ = 2 tan θ 1 − tan 2 θ tangent 2 theta equals . fraction 2 tangent theta , over 1 minus , tangent squared , theta end fraction

Half-Angle Identities

sin A 2 = ± 1 − cos A 2 cos A 2 = ± 1 + cos A 2 tan A 2 = ± 1 − cos A 1 + cos A table with 3 rows and 1 column , row1 column 1 , sine , eh over 2 , equals plus minus . square root of fraction 1 minus cosine eh , over 2 end fraction end root , row2 column 1 , cosine , eh over 2 , equals plus minus . square root of fraction 1 plus cosine eh , over 2 end fraction end root , row3 column 1 , tangent , eh over 2 , equals plus minus . square root of fraction 1 minus cosine eh , over 1 plus cosine eh end fraction end root , end table

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