Reference data and engineering information about natural trigonometric functions for mathematics applications.
Engineering reference data for Natural Trigonometric Functions in mathematics.
x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c
Roots of ax² + bx + c = 0.
c 2 = a 2 + b 2 c^2 = a^2 + b^2 c 2 = a 2 + b 2
Right triangle relationship.
A = π r 2 A = \pi r^2 A = π r 2
Area of a circle.
log b ( x ) = ln ( x ) ln ( b ) \log_b(x) = \frac{\ln(x)}{\ln(b)} log b ( x ) = l n ( b ) l n ( x )
Change of base formula.
Symbol Description Unit π \pi π Pi 3.14159... e e e Euler's number 2.71828...
The inverse trigonometric functions are defined as:
arcsin ( a ) = sin − 1 ( a ) ( 1 a ) \arcsin(a) = \sin^{-1}(a) \quad (1a) arcsin ( a ) = sin − 1 ( a ) ( 1 a ) arccos ( a ) = cos − 1 ( a ) ( 2 a ) \arccos(a) = \cos^{-1}(a) \quad (2a) arccos ( a ) = cos − 1 ( a ) ( 2 a ) arctan ( a ) = tan − 1 ( a ) ( 3 a ) \arctan(a) = \tan^{-1}(a) \quad (3a) arctan ( a ) = tan − 1 ( a ) ( 3 a ) sin ( a ± b ) = sin a cos b ± cos a sin b ( 5 ) \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \quad (5) sin ( a ± b ) = sin a cos b ± cos a sin b ( 5 ) cos ( a ± b ) = cos a cos b ∓ sin a sin b ( 5 b ) \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \quad (5b) cos ( a ± b ) = cos a cos b ∓ sin a sin b ( 5 b ) tan ( a ± b ) = tan a ± tan b 1 ∓ tan a tan b ( 5 c ) \tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b} \quad (5c) tan ( a ± b ) = 1 ∓ tan a tan b tan a ± tan b ( 5 c ) sin a + sin b = 2 sin ( a + b 2 ) cos ( a − b 2 ) ( 6 ) \sin a + \sin b = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right) \quad (6) sin a + sin b = 2 sin ( 2 a + b ) cos ( 2 a − b ) ( 6 ) sin a − sin b = 2 cos ( a + b 2 ) sin ( a − b 2 ) ( 6 b ) \sin a - \sin b = 2 \cos\left(\frac{a + b}{2}\right) \sin\left(\frac{a - b}{2}\right) \quad (6b) sin a − sin b = 2 cos ( 2 a + b ) sin ( 2 a − b ) ( 6 b ) cos a + cos b = 2 cos ( a + b 2 ) cos ( a − b 2 ) ( 6 c ) \cos a + \cos b = 2 \cos\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right) \quad (6c) cos a + cos b = 2 cos ( 2 a + b ) cos ( 2 a − b ) ( 6 c ) cos a − cos b = − 2 sin ( a + b 2 ) sin ( a − b 2 ) ( 6 d ) \cos a - \cos b = -2 \sin\left(\frac{a + b}{2}\right) \sin\left(\frac{a - b}{2}\right) \quad (6d) cos a − cos b = − 2 sin ( 2 a + b ) sin ( 2 a − b ) ( 6 d ) 2 sin a cos b = sin ( a − b ) + sin ( a + b ) ( 7 ) 2 \sin a \cos b = \sin(a - b) + \sin(a + b) \quad (7) 2 sin a cos b = sin ( a − b ) + sin ( a + b ) ( 7 ) 2 sin a sin b = cos ( a − b ) − cos ( a + b ) ( 7 b ) 2 \sin a \sin b = \cos(a - b) - \cos(a + b) \quad (7b) 2 sin a sin b = cos ( a − b ) − cos ( a + b ) ( 7 b ) 2 cos a cos b = cos ( a − b ) + cos ( a + b ) ( 7 c ) 2 \cos a \cos b = \cos(a - b) + \cos(a + b) \quad (7c) 2 cos a cos b = cos ( a − b ) + cos ( a + b ) ( 7 c ) sin ( 2 a ) = 2 sin a cos a ( 8 ) \sin(2a) = 2 \sin a \cos a \quad (8) sin ( 2 a ) = 2 sin a cos a ( 8 ) cos ( 2 a ) = cos 2 a − sin 2 a = 2 cos 2 a − 1 = 1 − 2 sin 2 a ( 8 b , 8 c , 8 d ) \cos(2a) = \cos^2 a - \sin^2 a = 2\cos^2 a - 1 = 1 - 2\sin^2 a \quad (8b, 8c, 8d) cos ( 2 a ) = cos 2 a − sin 2 a = 2 cos 2 a − 1 = 1 − 2 sin 2 a ( 8 b , 8 c , 8 d ) tan ( 2 a ) = 2 tan a 1 − tan 2 a ( 8 e ) \tan(2a) = \frac{2 \tan a}{1 - \tan^2 a} \quad (8e) tan ( 2 a ) = 1 − tan 2 a 2 tan a ( 8 e ) Fundamental Identity:
sin 2 a + cos 2 a = 1 ( 8 f ) \sin^2 a + \cos^2 a = 1 \quad (8f) sin 2 a + cos 2 a = 1 ( 8 f ) Cofunction Identities relate functions of complementary angles (sum to 90° or π/2 radians):
sin ( 90 ∘ − θ ) = cos θ , cos ( 90 ∘ − θ ) = sin θ ( 9 c , 10 c ) \sin(90^\circ - \theta) = \cos \theta, \quad \cos(90^\circ - \theta) = \sin \theta \quad (9c, 10c) sin ( 9 0 ∘ − θ ) = cos θ , cos ( 9 0 ∘ − θ ) = sin θ ( 9 c , 10 c ) tan ( 90 ∘ − θ ) = cot θ , cot ( 90 ∘ − θ ) = tan θ \tan(90^\circ - \theta) = \cot \theta, \quad \cot(90^\circ - \theta) = \tan \theta tan ( 9 0 ∘ − θ ) = cot θ , cot ( 9 0 ∘ − θ ) = tan θ Symmetry Properties:
Sine is an odd function: sin ( − θ ) = − sin ( θ ) \sin(-\theta) = -\sin(\theta) sin ( − θ ) = − sin ( θ ) Cosine is an even function: cos ( − θ ) = cos ( θ ) \cos(-\theta) = \cos(\theta) cos ( − θ ) = cos ( θ ) Tangent is an odd function: tan ( − θ ) = − tan ( θ ) \tan(-\theta) = -\tan(\theta) tan ( − θ ) = − tan ( θ )
Periodicity:
sin ( θ + 360 ∘ ) = sin θ , cos ( θ + 360 ∘ ) = cos θ , tan ( θ + 180 ∘ ) = tan θ \sin(\theta + 360^\circ) = \sin \theta, \quad \cos(\theta + 360^\circ) = \cos \theta, \quad \tan(\theta + 180^\circ) = \tan \theta sin ( θ + 36 0 ∘ ) = sin θ , cos ( θ + 36 0 ∘ ) = cos θ , tan ( θ + 18 0 ∘ ) = tan θ Angle Addition for Specific Angles:
The extracted text provides expanded relationships for functions of ( 90 ∘ ± θ ) (90^\circ \pm \theta) ( 9 0 ∘ ± θ ) ( 180 ∘ ± θ ) (180^\circ \pm \theta) ( 18 0 ∘ ± θ ) ( 270 ∘ ± θ ) (270^\circ \pm \theta) ( 27 0 ∘ ± θ ) ( 360 ∘ ± θ ) (360^\circ \pm \theta) ( 36 0 ∘ ± θ )
sin ( 90 ∘ + θ ) = cos θ , cos ( 90 ∘ + θ ) = − sin θ \sin(90^\circ + \theta) = \cos \theta, \quad \cos(90^\circ + \theta) = -\sin \theta sin ( 9 0 ∘ + θ ) = cos θ , cos ( 9 0 ∘ + θ ) = − sin θ sin ( 180 ∘ − θ ) = sin θ , cos ( 180 ∘ − θ ) = − cos θ \sin(180^\circ - \theta) = \sin \theta, \quad \cos(180^\circ - \theta) = -\cos \theta sin ( 18 0 ∘ − θ ) = sin θ , cos ( 18 0 ∘ − θ ) = − cos θ These are useful for simplifying expressions and solving equations.
